Rust f32::asinh

Percentage Accurate: 38.3% → 99.5%

Time: 10.1s

Alternatives: 22

Speedup: 4.0×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (asinh x))

C

float code(float x) {
	return asinhf(x);
}

Julia

function code(x)
	return asinh(x)
end

MATLAB

function tmp = code(x)
	tmp = asinh(x);
end

Initial Program: 38.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

C

float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
end

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + \mathsf{hypot}\left(1, x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log t\_1, x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right) - \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{t\_1}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (+ (fabs x) (hypot 1.0 x))))
   (if (<= t_0 -1.0)
     (copysign (log t_1) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (-
         (log1p
          (-
           (* (* x x) (+ 0.5 (* x (* x (+ -0.125 (* (* x x) 0.0625))))))
           (fabs x))))
        x)
       (copysign (- (log (/ 1.0 t_1))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = fabsf(x) + hypotf(1.0f, x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(logf(t_1), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(-log1pf((((x * x) * (0.5f + (x * (x * (-0.125f + ((x * x) * 0.0625f)))))) - fabsf(x))), x);
	} else {
		tmp = copysignf(-logf((1.0f / t_1)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = Float32(abs(x) + hypot(Float32(1.0), x))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(log(t_1), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-log1p(Float32(Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(x * Float32(x * Float32(Float32(-0.125) + Float32(Float32(x * x) * Float32(0.0625))))))) - abs(x)))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / t_1))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 23.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right)} \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right) \]

      3. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + {x}^{2}}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + x \cdot x}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{hypot}\left(1, x\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      8. fabs-lowering-fabs.f32 23.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   9. Simplified 23.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - \left|x\right|\right), x\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right), x\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) + -1 \cdot \left|x\right|\right)\right)\right), x\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(-1 \cdot \left|x\right| + \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(-1 \cdot \left|x\right| + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + -1 \cdot \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 - \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 - \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \left(\left|x\right|\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       10. fabs-lowering-fabs.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       11. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       13. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       14. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       17. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   12. Simplified 23.8%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(1 - \left|x\right|\right) + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)}, x\right) \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)\right), x\right)} \]
   14. Step-by-step derivation

       1. copysign-lowering-copysign.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

   15. Simplified 99.9%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right) - \left|x\right|\right), x\right)} \]

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{1}{\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}}\right), x\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right), x\right) \]

      4. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right), x\right) \]

      5. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}}\right)\right)\right), x\right) \]

   6. Applied egg-rr 98.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{1}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right) - \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))
   (if (<= t_0 -1.0)
     t_1
     (if (<= t_0 0.20000000298023224)
       (copysign
        (-
         (log1p
          (-
           (* (* x x) (+ 0.5 (* x (* x (+ -0.125 (* (* x x) 0.0625))))))
           (fabs x))))
        x)
       t_1))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = t_1;
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(-log1pf((((x * x) * (0.5f + (x * (x * (-0.125f + ((x * x) * 0.0625f)))))) - fabsf(x))), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = t_1;
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-log1p(Float32(Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(x * Float32(x * Float32(Float32(-0.125) + Float32(Float32(x * x) * Float32(0.0625))))))) - abs(x)))), x);
	else
		tmp = t_1;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -1 or 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 54.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 99.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 23.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right)} \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right) \]

      3. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + {x}^{2}}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + x \cdot x}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{hypot}\left(1, x\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      8. fabs-lowering-fabs.f32 23.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   9. Simplified 23.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - \left|x\right|\right), x\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right), x\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) + -1 \cdot \left|x\right|\right)\right)\right), x\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(-1 \cdot \left|x\right| + \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(-1 \cdot \left|x\right| + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + -1 \cdot \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 - \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 - \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \left(\left|x\right|\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       10. fabs-lowering-fabs.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       11. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       13. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       14. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       17. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   12. Simplified 23.8%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(1 - \left|x\right|\right) + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)}, x\right) \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)\right), x\right)} \]
   14. Step-by-step derivation

       1. copysign-lowering-copysign.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

   15. Simplified 99.9%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right) - \left|x\right|\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\frac{1}{\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right) - \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + \left(\frac{\left|x\right|}{x} + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign
    (log (/ 1.0 (/ 1.0 (+ (- (fabs x) x) (/ (+ -0.5 (/ 0.125 (* x x))) x)))))
    x)
   (if (<= x 0.20000000298023224)
     (copysign
      (-
       (log1p
        (-
         (* (* x x) (+ 0.5 (* x (* x (+ -0.125 (* (* x x) 0.0625))))))
         (fabs x))))
      x)
     (copysign
      (log
       (*
        x
        (+
         1.0
         (+
          (/ 0.5 (* x x))
          (+ (/ (fabs x) x) (/ -0.125 (* x (* x (* x x)))))))))
      x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((1.0f / (1.0f / ((fabsf(x) - x) + ((-0.5f + (0.125f / (x * x))) / x))))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-log1pf((((x * x) * (0.5f + (x * (x * (-0.125f + ((x * x) * 0.0625f)))))) - fabsf(x))), x);
	} else {
		tmp = copysignf(logf((x * (1.0f + ((0.5f / (x * x)) + ((fabsf(x) / x) + (-0.125f / (x * (x * (x * x))))))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(Float32(1.0) / Float32(Float32(abs(x) - x) + Float32(Float32(Float32(-0.5) + Float32(Float32(0.125) / Float32(x * x))) / x))))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-log1p(Float32(Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(x * Float32(x * Float32(Float32(-0.125) + Float32(Float32(x * x) * Float32(0.0625))))))) - abs(x)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(1.0) + Float32(Float32(Float32(0.5) / Float32(x * x)) + Float32(Float32(abs(x) / x) + Float32(Float32(-0.125) / Float32(x * Float32(x * Float32(x * x))))))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   7. Simplified 54.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \cdot x}{x} - x\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)} \]
   9. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{x}\right) \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right), x\right)} \]
   11. Step-by-step derivation

       1. flip-+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}\right)\right), x\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}}\right)\right), x\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}\right)\right)\right), x\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{1}{\frac{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}}\right)\right)\right), x\right) \]

       5. flip-+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{1}{\left(\left|x\right| - x\right) + \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}\right)\right)\right), x\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \left(\left(\left|x\right| - x\right) + \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}\right)\right)\right)\right), x\right) \]

       7. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\left|x\right| - x\right), \left(\frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}\right)\right)\right)\right)\right), x\right) \]

   12. Applied egg-rr 99.0%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{1}{\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}}\right)}, x\right) \]

   if -1 < x < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 23.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right)} \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right) \]

      3. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + {x}^{2}}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + x \cdot x}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{hypot}\left(1, x\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      8. fabs-lowering-fabs.f32 23.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   9. Simplified 23.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - \left|x\right|\right), x\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right), x\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) + -1 \cdot \left|x\right|\right)\right)\right), x\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(-1 \cdot \left|x\right| + \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(-1 \cdot \left|x\right| + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + -1 \cdot \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 - \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 - \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \left(\left|x\right|\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       10. fabs-lowering-fabs.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       11. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       13. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       14. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       17. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   12. Simplified 23.8%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(1 - \left|x\right|\right) + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)}, x\right) \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)\right), x\right)} \]
   14. Step-by-step derivation

       1. copysign-lowering-copysign.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

   15. Simplified 99.9%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right) - \left|x\right|\right), x\right)} \]

   if 0.200000003 < x

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right) - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right) - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right), x\right) \]

      2. associate--l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(1 + \left(\left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}} + \left(\frac{\left|x\right|}{x} - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2}}{{x}^{2}}\right), \left(\frac{\left|x\right|}{x} - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right), \left(\frac{\left|x\right|}{x} - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right), \left(\frac{\left|x\right|}{x} - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \left(\frac{\left|x\right|}{x} - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \left(\frac{\left|x\right|}{x} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      10. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\left(\frac{\left|x\right|}{x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), x\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      12. fabs-lowering-fabs.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{8}}{{x}^{\left(2 \cdot 2\right)}}\right)\right)\right)\right)\right)\right), x\right) \]

      18. pow-sqr N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{8}}{{x}^{2} \cdot {x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      19. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

   7. Simplified 96.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + \left(\frac{\left|x\right|}{x} + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\frac{1}{\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right) - \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign
    (log (/ 1.0 (/ 1.0 (+ (- (fabs x) x) (/ (+ -0.5 (/ 0.125 (* x x))) x)))))
    x)
   (if (<= x 0.20000000298023224)
     (copysign
      (-
       (log1p
        (-
         (* (* x x) (+ 0.5 (* x (* x (+ -0.125 (* (* x x) 0.0625))))))
         (fabs x))))
      x)
     (copysign
      (- (log (/ 1.0 (* x (+ (/ (fabs x) x) (+ 1.0 (/ 0.5 (* x x))))))))
      x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((1.0f / (1.0f / ((fabsf(x) - x) + ((-0.5f + (0.125f / (x * x))) / x))))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-log1pf((((x * x) * (0.5f + (x * (x * (-0.125f + ((x * x) * 0.0625f)))))) - fabsf(x))), x);
	} else {
		tmp = copysignf(-logf((1.0f / (x * ((fabsf(x) / x) + (1.0f + (0.5f / (x * x))))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(Float32(1.0) / Float32(Float32(abs(x) - x) + Float32(Float32(Float32(-0.5) + Float32(Float32(0.125) / Float32(x * x))) / x))))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-log1p(Float32(Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(x * Float32(x * Float32(Float32(-0.125) + Float32(Float32(x * x) * Float32(0.0625))))))) - abs(x)))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / Float32(x * Float32(Float32(abs(x) / x) + Float32(Float32(1.0) + Float32(Float32(0.5) / Float32(x * x)))))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   7. Simplified 54.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \cdot x}{x} - x\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)} \]
   9. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{x}\right) \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right), x\right)} \]
   11. Step-by-step derivation

       1. flip-+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}\right)\right), x\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}}\right)\right), x\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}\right)\right)\right), x\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{1}{\frac{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}}\right)\right)\right), x\right) \]

       5. flip-+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{1}{\left(\left|x\right| - x\right) + \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}\right)\right)\right), x\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \left(\left(\left|x\right| - x\right) + \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}\right)\right)\right)\right), x\right) \]

       7. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\left|x\right| - x\right), \left(\frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}\right)\right)\right)\right)\right), x\right) \]

   12. Applied egg-rr 99.0%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{1}{\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}}\right)}, x\right) \]

   if -1 < x < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 23.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right)} \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right) \]

      3. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + {x}^{2}}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + x \cdot x}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{hypot}\left(1, x\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      8. fabs-lowering-fabs.f32 23.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   9. Simplified 23.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - \left|x\right|\right), x\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right), x\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) + -1 \cdot \left|x\right|\right)\right)\right), x\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(-1 \cdot \left|x\right| + \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(-1 \cdot \left|x\right| + 1\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + -1 \cdot \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 - \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 - \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \left(\left|x\right|\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       10. fabs-lowering-fabs.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       11. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       13. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

       14. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       17. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   12. Simplified 23.8%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(1 - \left|x\right|\right) + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)}, x\right) \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)\right), x\right)} \]
   14. Step-by-step derivation

       1. copysign-lowering-copysign.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right) - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

   15. Simplified 99.9%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right) - \left|x\right|\right), x\right)} \]

   if 0.200000003 < x

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{1}{\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}}\right), x\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right), x\right) \]

      4. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right), x\right) \]

      5. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}}\right)\right)\right), x\right) \]

   6. Applied egg-rr 98.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{1}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{\left(x \cdot \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)}\right)\right)\right), x\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right)\right)\right), x\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) + \frac{\left|x\right|}{x}\right)\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\frac{\left|x\right|}{x} + \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\left|x\right|}{x}\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), x\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      6. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 95.8%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   9. Simplified 95.8%

      \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{1}{\color{blue}{x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)}}\right), x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\frac{1}{\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot 0.5 - \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign
    (log (/ 1.0 (/ 1.0 (+ (- (fabs x) x) (/ (+ -0.5 (/ 0.125 (* x x))) x)))))
    x)
   (if (<= x 0.20000000298023224)
     (copysign (- (log1p (- (* (* x x) 0.5) (fabs x)))) x)
     (copysign
      (- (log (/ 1.0 (* x (+ (/ (fabs x) x) (+ 1.0 (/ 0.5 (* x x))))))))
      x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((1.0f / (1.0f / ((fabsf(x) - x) + ((-0.5f + (0.125f / (x * x))) / x))))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-log1pf((((x * x) * 0.5f) - fabsf(x))), x);
	} else {
		tmp = copysignf(-logf((1.0f / (x * ((fabsf(x) / x) + (1.0f + (0.5f / (x * x))))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(Float32(1.0) / Float32(Float32(abs(x) - x) + Float32(Float32(Float32(-0.5) + Float32(Float32(0.125) / Float32(x * x))) / x))))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-log1p(Float32(Float32(Float32(x * x) * Float32(0.5)) - abs(x)))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / Float32(x * Float32(Float32(abs(x) / x) + Float32(Float32(1.0) + Float32(Float32(0.5) / Float32(x * x)))))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   7. Simplified 54.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \cdot x}{x} - x\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)} \]
   9. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{x}\right) \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right), x\right)} \]
   11. Step-by-step derivation

       1. flip-+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}\right)\right), x\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}}\right)\right), x\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}\right)\right)\right), x\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{1}{\frac{\left(\left|x\right| - x\right) \cdot \left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}{\left(\left|x\right| - x\right) - \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}}\right)\right)\right), x\right) \]

       5. flip-+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{1}{\left(\left|x\right| - x\right) + \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}}\right)\right)\right), x\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \left(\left(\left|x\right| - x\right) + \frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}\right)\right)\right)\right), x\right) \]

       7. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\left|x\right| - x\right), \left(\frac{\frac{-1}{2} + \frac{\frac{1}{8}}{x \cdot x}}{x}\right)\right)\right)\right)\right), x\right) \]

   12. Applied egg-rr 99.0%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{1}{\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}}\right)}, x\right) \]

   if -1 < x < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 23.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right)} \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right) \]

      3. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + {x}^{2}}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + x \cdot x}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{hypot}\left(1, x\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      8. fabs-lowering-fabs.f32 23.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   9. Simplified 23.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - \left|x\right|\right), x\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right), x\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) + -1 \cdot \left|x\right|\right)\right)\right), x\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(-1 \cdot \left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(-1 \cdot \left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + -1 \cdot \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 - \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 - \left|x\right|\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \left(\left|x\right|\right)\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       10. fabs-lowering-fabs.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

       14. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

       15. *-lowering-*.f32 22.9%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   12. Simplified 22.9%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(1 - \left|x\right|\right) + x \cdot \left(x \cdot 0.5\right)\right)}, x\right) \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right), x\right)} \]
   14. Step-by-step derivation

       1. copysign-lowering-copysign.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

       2. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right), x\right) \]

       3. associate--l+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(1 + \left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       4. log1p-define N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\left(\mathsf{log1p}\left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       5. log1p-lowering-log1p.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       6. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\left(\frac{1}{2} \cdot {x}^{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\left({x}^{2} \cdot \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       11. fabs-lowering-fabs.f32 98.9%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   15. Simplified 98.9%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot 0.5 - \left|x\right|\right), x\right)} \]

   if 0.200000003 < x

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{1}{\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}}\right), x\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right), x\right) \]

      4. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right), x\right) \]

      5. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}}\right)\right)\right), x\right) \]

   6. Applied egg-rr 98.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{1}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{\left(x \cdot \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)}\right)\right)\right), x\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right)\right)\right), x\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) + \frac{\left|x\right|}{x}\right)\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\frac{\left|x\right|}{x} + \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\left|x\right|}{x}\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), x\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      6. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 95.8%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   9. Simplified 95.8%

      \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{1}{\color{blue}{x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)}}\right), x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot 0.5 - \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (+ (- (fabs x) x) (/ (+ -0.5 (/ 0.125 (* x x))) x))) x)
   (if (<= x 0.20000000298023224)
     (copysign (- (log1p (- (* (* x x) 0.5) (fabs x)))) x)
     (copysign
      (- (log (/ 1.0 (* x (+ (/ (fabs x) x) (+ 1.0 (/ 0.5 (* x x))))))))
      x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + ((-0.5f + (0.125f / (x * x))) / x))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-log1pf((((x * x) * 0.5f) - fabsf(x))), x);
	} else {
		tmp = copysignf(-logf((1.0f / (x * ((fabsf(x) / x) + (1.0f + (0.5f / (x * x))))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(Float32(-0.5) + Float32(Float32(0.125) / Float32(x * x))) / x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-log1p(Float32(Float32(Float32(x * x) * Float32(0.5)) - abs(x)))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / Float32(x * Float32(Float32(abs(x) / x) + Float32(Float32(1.0) + Float32(Float32(0.5) / Float32(x * x)))))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   7. Simplified 54.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \cdot x}{x} - x\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)} \]
   9. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{x}\right) \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right), x\right)} \]

   if -1 < x < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 23.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right)} \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right) \]

      3. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + {x}^{2}}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + x \cdot x}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{hypot}\left(1, x\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      8. fabs-lowering-fabs.f32 23.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   9. Simplified 23.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - \left|x\right|\right), x\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right), x\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) + -1 \cdot \left|x\right|\right)\right)\right), x\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(-1 \cdot \left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(-1 \cdot \left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + -1 \cdot \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 - \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 - \left|x\right|\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \left(\left|x\right|\right)\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       10. fabs-lowering-fabs.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

       14. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

       15. *-lowering-*.f32 22.9%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   12. Simplified 22.9%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(1 - \left|x\right|\right) + x \cdot \left(x \cdot 0.5\right)\right)}, x\right) \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right), x\right)} \]
   14. Step-by-step derivation

       1. copysign-lowering-copysign.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

       2. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right), x\right) \]

       3. associate--l+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(1 + \left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       4. log1p-define N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\left(\mathsf{log1p}\left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       5. log1p-lowering-log1p.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       6. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\left(\frac{1}{2} \cdot {x}^{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\left({x}^{2} \cdot \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       11. fabs-lowering-fabs.f32 98.9%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   15. Simplified 98.9%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot 0.5 - \left|x\right|\right), x\right)} \]

   if 0.200000003 < x

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{1}{\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}}\right), x\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right), x\right) \]

      4. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right), x\right) \]

      5. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}}\right)\right)\right), x\right) \]

   6. Applied egg-rr 98.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{1}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{\left(x \cdot \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)}\right)\right)\right), x\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right)\right)\right), x\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) + \frac{\left|x\right|}{x}\right)\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\frac{\left|x\right|}{x} + \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\left|x\right|}{x}\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), x\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      6. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right), x\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 95.8%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   9. Simplified 95.8%

      \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{1}{\color{blue}{x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)}}\right), x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot 0.5 - \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (+ (- (fabs x) x) (/ (+ -0.5 (/ 0.125 (* x x))) x))) x)
   (if (<= x 0.20000000298023224)
     (copysign (- (log1p (- (* (* x x) 0.5) (fabs x)))) x)
     (copysign (log (* x (+ (/ (fabs x) x) (+ 1.0 (/ 0.5 (* x x)))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + ((-0.5f + (0.125f / (x * x))) / x))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-log1pf((((x * x) * 0.5f) - fabsf(x))), x);
	} else {
		tmp = copysignf(logf((x * ((fabsf(x) / x) + (1.0f + (0.5f / (x * x)))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(Float32(-0.5) + Float32(Float32(0.125) / Float32(x * x))) / x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-log1p(Float32(Float32(Float32(x * x) * Float32(0.5)) - abs(x)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(abs(x) / x) + Float32(Float32(1.0) + Float32(Float32(0.5) / Float32(x * x)))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   7. Simplified 54.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \cdot x}{x} - x\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)} \]
   9. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{x}\right) \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right), x\right)} \]

   if -1 < x < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 23.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right)} \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right) \]

      3. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + {x}^{2}}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + x \cdot x}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{hypot}\left(1, x\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      8. fabs-lowering-fabs.f32 23.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   9. Simplified 23.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - \left|x\right|\right), x\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right), x\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) + -1 \cdot \left|x\right|\right)\right)\right), x\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(-1 \cdot \left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(-1 \cdot \left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + -1 \cdot \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 - \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 - \left|x\right|\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \left(\left|x\right|\right)\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       10. fabs-lowering-fabs.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

       14. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

       15. *-lowering-*.f32 22.9%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   12. Simplified 22.9%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(1 - \left|x\right|\right) + x \cdot \left(x \cdot 0.5\right)\right)}, x\right) \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right), x\right)} \]
   14. Step-by-step derivation

       1. copysign-lowering-copysign.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

       2. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right), x\right) \]

       3. associate--l+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(1 + \left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       4. log1p-define N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\left(\mathsf{log1p}\left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       5. log1p-lowering-log1p.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       6. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\left(\frac{1}{2} \cdot {x}^{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\left({x}^{2} \cdot \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       11. fabs-lowering-fabs.f32 98.9%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   15. Simplified 98.9%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot 0.5 - \left|x\right|\right), x\right)} \]

   if 0.200000003 < x

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) + \frac{\left|x\right|}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\left|x\right|}{x} + \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\left|x\right|}{x}\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), x\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      6. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 95.8%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   7. Simplified 95.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot 0.5 - \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
   (if (<= x 0.20000000298023224)
     (copysign (- (log1p (- (* (* x x) 0.5) (fabs x)))) x)
     (copysign (log (* x (+ (/ (fabs x) x) (+ 1.0 (/ 0.5 (* x x)))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-log1pf((((x * x) * 0.5f) - fabsf(x))), x);
	} else {
		tmp = copysignf(logf((x * ((fabsf(x) / x) + (1.0f + (0.5f / (x * x)))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-log1p(Float32(Float32(Float32(x * x) * Float32(0.5)) - abs(x)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(abs(x) / x) + Float32(Float32(1.0) + Float32(Float32(0.5) / Float32(x * x)))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   7. Simplified 54.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \cdot x}{x} - x\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)} \]
   9. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{x}\right) \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right), x\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right)\right), x\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot 1}{x}\right)\right)\right), x\right) \]

       2. *-inverses N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{{x}^{2}}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{x \cdot x}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       4. sqr-abs N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{\left|x\right| \cdot \left|x\right|}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{\left|x\right| \cdot \left|x\right|}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       8. sqr-abs N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{x \cdot x}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       10. *-inverses N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot 1\right), x\right)\right)\right), x\right) \]

       11. metadata-eval 98.3%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right)\right), x\right) \]

   13. Simplified 98.3%

       \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \color{blue}{\frac{-0.5}{x}}\right), x\right) \]

   if -1 < x < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 23.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right)} \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right) \]

      3. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + {x}^{2}}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + x \cdot x}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{hypot}\left(1, x\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      8. fabs-lowering-fabs.f32 23.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   9. Simplified 23.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - \left|x\right|\right), x\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right), x\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) + -1 \cdot \left|x\right|\right)\right)\right), x\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(-1 \cdot \left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(-1 \cdot \left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + -1 \cdot \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 - \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 - \left|x\right|\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \left(\left|x\right|\right)\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       10. fabs-lowering-fabs.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

       14. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

       15. *-lowering-*.f32 22.9%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   12. Simplified 22.9%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(1 - \left|x\right|\right) + x \cdot \left(x \cdot 0.5\right)\right)}, x\right) \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right), x\right)} \]
   14. Step-by-step derivation

       1. copysign-lowering-copysign.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

       2. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right), x\right) \]

       3. associate--l+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(1 + \left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       4. log1p-define N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\left(\mathsf{log1p}\left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       5. log1p-lowering-log1p.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       6. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\left(\frac{1}{2} \cdot {x}^{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\left({x}^{2} \cdot \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       11. fabs-lowering-fabs.f32 98.9%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   15. Simplified 98.9%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot 0.5 - \left|x\right|\right), x\right)} \]

   if 0.200000003 < x

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(\left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right) + \frac{\left|x\right|}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\left|x\right|}{x} + \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\left|x\right|}{x}\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), x\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      6. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(1 + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      7. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 95.8%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   7. Simplified 95.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot 0.5 - \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
   (if (<= x 0.20000000298023224)
     (copysign (- (log1p (- (* (* x x) 0.5) (fabs x)))) x)
     (copysign (log (* x (+ 1.0 (/ (fabs x) x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-log1pf((((x * x) * 0.5f) - fabsf(x))), x);
	} else {
		tmp = copysignf(logf((x * (1.0f + (fabsf(x) / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-log1p(Float32(Float32(Float32(x * x) * Float32(0.5)) - abs(x)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(1.0) + Float32(abs(x) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   7. Simplified 54.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \cdot x}{x} - x\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)} \]
   9. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{x}\right) \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right), x\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right)\right), x\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot 1}{x}\right)\right)\right), x\right) \]

       2. *-inverses N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{{x}^{2}}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{x \cdot x}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       4. sqr-abs N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{\left|x\right| \cdot \left|x\right|}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{\left|x\right| \cdot \left|x\right|}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       8. sqr-abs N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{x \cdot x}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       10. *-inverses N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot 1\right), x\right)\right)\right), x\right) \]

       11. metadata-eval 98.3%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right)\right), x\right) \]

   13. Simplified 98.3%

       \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \color{blue}{\frac{-0.5}{x}}\right), x\right) \]

   if -1 < x < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 23.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right)} \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right), x\right) \]

      3. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\sqrt{1 + {x}^{2}} - \left|x\right|\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + {x}^{2}}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{1 + x \cdot x}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{hypot}\left(1, x\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

      8. fabs-lowering-fabs.f32 23.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{hypot.f32}\left(1, x\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   9. Simplified 23.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - \left|x\right|\right), x\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right)\right)\right), x\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) + -1 \cdot \left|x\right|\right)\right)\right), x\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(-1 \cdot \left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       4. associate-+r+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(-1 \cdot \left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + -1 \cdot \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(\left|x\right|\right)\right)\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\left(1 - \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 - \left|x\right|\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \left(\left|x\right|\right)\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       10. fabs-lowering-fabs.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

       14. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

       15. *-lowering-*.f32 22.9%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   12. Simplified 22.9%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(1 - \left|x\right|\right) + x \cdot \left(x \cdot 0.5\right)\right)}, x\right) \]

   13. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right), x\right)} \]
   14. Step-by-step derivation

       1. copysign-lowering-copysign.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right)\right), \color{blue}{x}\right) \]

       2. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\left(1 + \frac{1}{2} \cdot {x}^{2}\right) - \left|x\right|\right)\right), x\right) \]

       3. associate--l+ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(1 + \left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       4. log1p-define N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\left(\mathsf{log1p}\left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       5. log1p-lowering-log1p.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{1}{2} \cdot {x}^{2} - \left|x\right|\right)\right)\right), x\right) \]

       6. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\left(\frac{1}{2} \cdot {x}^{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\left({x}^{2} \cdot \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

       11. fabs-lowering-fabs.f32 98.9%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \mathsf{fabs.f32}\left(x\right)\right)\right)\right), x\right) \]

   15. Simplified 98.9%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(x \cdot x\right) \cdot 0.5 - \left|x\right|\right), x\right)} \]

   if 0.200000003 < x

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(1 + \frac{\left|x\right|}{x}\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(\left|x\right|\right), x\right)\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 93.5%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right)\right)\right)\right), x\right) \]

   7. Simplified 93.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 97.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
   (if (<= x 2.0)
     (copysign x x)
     (copysign (log (* x (+ 1.0 (/ (fabs x) x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * (1.0f + (fabsf(x) / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(1.0) + Float32(abs(x) / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(log(((abs(x) - x) + (single(-0.5) / x))));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * (single(1.0) + (abs(x) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   7. Simplified 54.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \cdot x}{x} - x\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right)} \]
   9. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right) - \left(x + \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{x}\right) \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right), x\right)} \]

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right)\right), x\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot 1}{x}\right)\right)\right), x\right) \]

       2. *-inverses N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{{x}^{2}}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{x \cdot x}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       4. sqr-abs N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{\left|x\right| \cdot \left|x\right|}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{-1}{2} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{x}^{2}}}{x}\right)\right)\right), x\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{\left|x\right| \cdot \left|x\right|}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       8. sqr-abs N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{x \cdot x}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

       10. *-inverses N/A

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot 1\right), x\right)\right)\right), x\right) \]

       11. metadata-eval 98.3%

           \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right)\right), x\right) \]

   13. Simplified 98.3%

       \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \color{blue}{\frac{-0.5}{x}}\right), x\right) \]

   if -1 < x < 2

   1. Initial program 23.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.5%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.5%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 92.3%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 92.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 15.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Simplified 97.3%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 2 < x

   1. Initial program 53.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(1 + \frac{\left|x\right|}{x}\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(\left|x\right|\right), x\right)\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 94.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right)\right)\right)\right), x\right) \]

   7. Simplified 94.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 2.0)
     (copysign x x)
     (copysign (log (* x (+ 1.0 (/ (fabs x) x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * (1.0f + (fabsf(x) / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(1.0) + Float32(abs(x) / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(log((abs(x) - x)));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * (single(1.0) + (abs(x) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right|}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right) + x \cdot 1\right)\right)\right), x\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right) + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right)\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right)\right)\right) - x\right)\right), x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(\mathsf{neg}\left(\frac{\left|x\right|}{x}\right)\right)\right)\right) - x\right)\right), x\right) \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot \frac{\left|x\right|}{x}\right)\right)\right)\right) - x\right)\right), x\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \frac{\left|x\right|}{x} - x\right)\right), x\right) \]

      11. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(x \cdot \frac{\left|x\right|}{x}\right), x\right)\right), x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\left|x\right|}{x} \cdot x\right), x\right)\right), x\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\left|x\right| \cdot x}{x}\right), x\right)\right), x\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot x\right), x\right), x\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(x \cdot \left|x\right|\right), x\right), x\right)\right), x\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left(\left|x\right|\right)\right), x\right), x\right)\right), x\right) \]

      17. fabs-lowering-fabs.f32 51.7%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{fabs.f32}\left(x\right)\right), x\right), x\right)\right), x\right) \]

   7. Simplified 51.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{x \cdot \left|x\right|}{x} - x\right)}, x\right) \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{x \cdot \left|x\right|}{x} - x\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{x \cdot \left|x\right|}{x} - x\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left|x\right| \cdot x}{x} - x\right)\right), x\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| \cdot \frac{x}{x} - x\right)\right), x\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| \cdot 1 - x\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(1 \cdot \left|x\right| - x\right)\right), x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\left|x\right|\right), x\right)\right), x\right) \]

      9. fabs-lowering-fabs.f32 96.4%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right)\right), x\right) \]

   9. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)} \]

   if -1 < x < 2

   1. Initial program 23.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.5%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.5%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 92.3%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 92.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 15.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Simplified 97.3%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 2 < x

   1. Initial program 53.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \left(1 + \frac{\left|x\right|}{x}\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(\left|x\right|\right), x\right)\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 94.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right)\right)\right)\right), x\right) \]

   7. Simplified 94.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\left|x\right|}{x} + \log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 0.20000000298023224)
     (copysign x x)
     (copysign (+ (/ (fabs x) x) (log x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(((fabsf(x) / x) + logf(x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(Float32(abs(x) / x) + log(x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(log((abs(x) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(((abs(x) / x) + log(x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right|}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right) + x \cdot 1\right)\right)\right), x\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right) + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right)\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right)\right)\right) - x\right)\right), x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(\mathsf{neg}\left(\frac{\left|x\right|}{x}\right)\right)\right)\right) - x\right)\right), x\right) \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot \frac{\left|x\right|}{x}\right)\right)\right)\right) - x\right)\right), x\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \frac{\left|x\right|}{x} - x\right)\right), x\right) \]

      11. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(x \cdot \frac{\left|x\right|}{x}\right), x\right)\right), x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\left|x\right|}{x} \cdot x\right), x\right)\right), x\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\left|x\right| \cdot x}{x}\right), x\right)\right), x\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot x\right), x\right), x\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(x \cdot \left|x\right|\right), x\right), x\right)\right), x\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left(\left|x\right|\right)\right), x\right), x\right)\right), x\right) \]

      17. fabs-lowering-fabs.f32 51.7%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{fabs.f32}\left(x\right)\right), x\right), x\right)\right), x\right) \]

   7. Simplified 51.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{x \cdot \left|x\right|}{x} - x\right)}, x\right) \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{x \cdot \left|x\right|}{x} - x\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{x \cdot \left|x\right|}{x} - x\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left|x\right| \cdot x}{x} - x\right)\right), x\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| \cdot \frac{x}{x} - x\right)\right), x\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| \cdot 1 - x\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(1 \cdot \left|x\right| - x\right)\right), x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\left|x\right|\right), x\right)\right), x\right) \]

      9. fabs-lowering-fabs.f32 96.4%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right)\right), x\right) \]

   9. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)} \]

   if -1 < x < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 92.8%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 92.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 15.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Simplified 97.8%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.200000003 < x

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \frac{\left|x\right|}{x}\right)}, x\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\left(-1 \cdot \log \left(\frac{1}{x}\right)\right), \left(\frac{\left|x\right|}{x}\right)\right), x\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), \left(\frac{\left|x\right|}{x}\right)\right), x\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), \left(\frac{\left|x\right|}{x}\right)\right), x\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\log x, \left(\frac{\left|x\right|}{x}\right)\right), x\right) \]

      5. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\frac{\left|x\right|}{x}\right)\right), x\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\left(\left|x\right|\right), x\right)\right), x\right) \]

      7. fabs-lowering-fabs.f32 47.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right)\right), x\right) \]

   7. Simplified 47.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x + \frac{\left|x\right|}{x}}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\left|x\right|}{x} + \log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (- (fabs x) x)) x)
   (copysign (log1p (fabs x)) x)))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else {
		tmp = copysignf(log1pf(fabsf(x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	else
		tmp = copysign(log1p(abs(x)), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right|}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right) + x \cdot 1\right)\right)\right), x\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right) + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right)\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{\left|x\right|}{x}\right)\right)\right) - x\right)\right), x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(\mathsf{neg}\left(\frac{\left|x\right|}{x}\right)\right)\right)\right) - x\right)\right), x\right) \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot \frac{\left|x\right|}{x}\right)\right)\right)\right) - x\right)\right), x\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \frac{\left|x\right|}{x} - x\right)\right), x\right) \]

      11. --lowering--.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(x \cdot \frac{\left|x\right|}{x}\right), x\right)\right), x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\left|x\right|}{x} \cdot x\right), x\right)\right), x\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\left|x\right| \cdot x}{x}\right), x\right)\right), x\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot x\right), x\right), x\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(x \cdot \left|x\right|\right), x\right), x\right)\right), x\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left(\left|x\right|\right)\right), x\right), x\right)\right), x\right) \]

      17. fabs-lowering-fabs.f32 51.7%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{fabs.f32}\left(x\right)\right), x\right), x\right)\right), x\right) \]

   7. Simplified 51.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{x \cdot \left|x\right|}{x} - x\right)}, x\right) \]
   8. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{x \cdot \left|x\right|}{x} - x\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{x \cdot \left|x\right|}{x} - x\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left|x\right| \cdot x}{x} - x\right)\right), x\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| \cdot \frac{x}{x} - x\right)\right), x\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| \cdot 1 - x\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(1 \cdot \left|x\right| - x\right)\right), x\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\left|x\right|\right), x\right)\right), x\right) \]

      9. fabs-lowering-fabs.f32 96.4%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right)\right), x\right) \]

   9. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)} \]

   if -1 < x

   1. Initial program 32.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 45.1%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 45.1%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 78.4%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 78.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 68.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (log1p (fabs x)) x))

C

float code(float x) {
	return copysignf(log1pf(fabsf(x)), x);
}

Julia

function code(x)
	return copysign(log1p(abs(x)), x)
end

Derivation

1. Initial program 38.3%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Step-by-step derivation

   1. copysign-lowering-copysign.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

   2. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

   4. fabs-lowering-fabs.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

   6. hypot-1-def N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   7. hypot-lowering-hypot.f32 59.9%

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

3. Simplified 59.9%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
6. Step-by-step derivation

   1. log1p-define N/A

      \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

   2. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

   3. fabs-lowering-fabs.f32 69.0%

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

7. Simplified 69.0%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]
8. Add Preprocessing

Alternative 15: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50000000188416:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125 + \left(x \cdot x\right) \cdot -0.5}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \frac{1}{x}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50000000188416.0)
   (copysign (log (/ -1.0 x)) x)
   (if (<= x -1.0)
     (copysign (log (/ (+ 0.125 (* (* x x) -0.5)) (* x (* x x)))) x)
     (if (<= x 0.20000000298023224)
       (copysign x x)
       (copysign (+ (log x) (/ 1.0 x)) x)))))

C

float code(float x) {
	float tmp;
	if (x <= -50000000188416.0f) {
		tmp = copysignf(logf((-1.0f / x)), x);
	} else if (x <= -1.0f) {
		tmp = copysignf(logf(((0.125f + ((x * x) * -0.5f)) / (x * (x * x)))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf((logf(x) + (1.0f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50000000188416.0))
		tmp = copysign(log(Float32(Float32(-1.0) / x)), x);
	elseif (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(Float32(0.125) + Float32(Float32(x * x) * Float32(-0.5))) / Float32(x * Float32(x * x)))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(log(x) + Float32(Float32(1.0) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-50000000188416.0))
		tmp = sign(x) * abs(log((single(-1.0) / x)));
	elseif (x <= single(-1.0))
		tmp = sign(x) * abs(log(((single(0.125) + ((x * x) * single(-0.5))) / (x * (x * x)))));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs((log(x) + (single(1.0) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if x < -5.00000002e13

   1. Initial program 31.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(\frac{-1}{x}\right)}, x\right) \]
   8. Step-by-step derivation

      1. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{-1}{x}\right)\right), x\right) \]

      2. /-lowering-/.f32 46.3%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(-1, x\right)\right), x\right) \]

   9. Simplified 46.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\frac{-1}{x}\right)}, x\right) \]

   if -5.00000002e13 < x < -1

   1. Initial program 100.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   7. Simplified 97.1%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \cdot x}{x} - x\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{8} + \frac{-1}{2} \cdot {x}^{2}}{{x}^{3}}\right)}\right), x\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{8} + \frac{-1}{2} \cdot {x}^{2}\right), \left({x}^{3}\right)\right)\right), x\right) \]

   10. Simplified 94.7%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.125 + -0.5 \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot x\right)}\right)}, x\right) \]

   if -1 < x < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 92.8%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 92.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 15.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Simplified 97.8%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.200000003 < x

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 43.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 43.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 8.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(\frac{1}{x} - \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\frac{1}{x} + \left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)\right), x\right) \]

       2. log-rec N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\frac{1}{x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right)\right), x\right) \]

       3. remove-double-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\frac{1}{x} + \log x\right), x\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{x}\right), \log x\right), x\right) \]

       5. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, x\right), \log x\right), x\right) \]

       6. log-lowering-log.f32 44.0%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, x\right), \mathsf{log.f32}\left(x\right)\right), x\right) \]

   12. Simplified 44.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{1}{x} + \log x}, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50000000188416:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125 + \left(x \cdot x\right) \cdot -0.5}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \frac{1}{x}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \frac{1}{x}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (- (log (- 1.0 x))) x)
   (if (<= x 0.20000000298023224)
     (copysign x x)
     (copysign (+ (log x) (/ 1.0 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(-logf((1.0f - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf((logf(x) + (1.0f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) - x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(log(x) + Float32(Float32(1.0) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(-log((single(1.0) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs((log(x) + (single(1.0) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 43.8%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 43.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + -1 \cdot x\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), x\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 - x\right)\right)\right), x\right) \]

       3. --lowering--.f32 43.8%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, x\right)\right)\right), x\right) \]

   12. Simplified 43.8%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(1 - x\right)}, x\right) \]

   if -1 < x < 0.200000003

   1. Initial program 22.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 92.8%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 92.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 15.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Simplified 97.8%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.200000003 < x

   1. Initial program 54.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 43.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 43.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 8.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(\frac{1}{x} - \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\frac{1}{x} + \left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)\right), x\right) \]

       2. log-rec N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\frac{1}{x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right)\right), x\right) \]

       3. remove-double-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\left(\frac{1}{x} + \log x\right), x\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{x}\right), \log x\right), x\right) \]

       5. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, x\right), \log x\right), x\right) \]

       6. log-lowering-log.f32 44.0%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, x\right), \mathsf{log.f32}\left(x\right)\right), x\right) \]

   12. Simplified 44.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{1}{x} + \log x}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \frac{1}{x}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (- (log (- 1.0 x))) x)
   (if (<= x 2.0) (copysign x x) (copysign (- (log (/ 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(-logf((1.0f - x)), x);
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(-logf((1.0f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) - x))), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(-log((single(1.0) - x)));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(-log((single(1.0) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 43.8%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 43.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + -1 \cdot x\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), x\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 - x\right)\right)\right), x\right) \]

       3. --lowering--.f32 43.8%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, x\right)\right)\right), x\right) \]

   12. Simplified 43.8%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(1 - x\right)}, x\right) \]

   if -1 < x < 2

   1. Initial program 23.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.5%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.5%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 92.3%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 92.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 15.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Simplified 97.3%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 2 < x

   1. Initial program 53.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{1}{\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}}\right), x\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right), x\right) \]

      4. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right), x\right) \]

      5. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}}\right)\right)\right), x\right) \]

   6. Applied egg-rr 98.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{1}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{1}{x}\right)}\right)\right), x\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, x\right)\right)\right), x\right) \]

   9. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{1}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (- (log (- 1.0 x))) x)
   (if (<= x 2.0) (copysign x x) (copysign (log x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(-logf((1.0f - x)), x);
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) - x))), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(-log((single(1.0) - x)));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 43.8%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 43.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + -1 \cdot x\right)}\right)\right), x\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), x\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 - x\right)\right)\right), x\right) \]

       3. --lowering--.f32 43.8%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, x\right)\right)\right), x\right) \]

   12. Simplified 43.8%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(1 - x\right)}, x\right) \]

   if -1 < x < 2

   1. Initial program 23.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 23.5%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 23.5%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 92.3%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 92.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 15.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Simplified 97.3%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 2 < x

   1. Initial program 53.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log x, x\right) \]

      4. log-lowering-log.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   7. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -1.0 x)) x)
   (if (<= x 2.0) (copysign x x) (copysign (log x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-1.0f / x)), x);
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-1.0) / x)), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-1.0) / x)));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 54.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}{\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)}\right), x\right) \]

      3. log-div N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) - \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{\_.f32}\left(\log \left(\mathsf{neg}\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right), \log \left(\mathsf{neg}\left(\left(\left|x\right| - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 8.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(-\left(x \cdot x - \left(1 + x \cdot x\right)\right)\right) - \log \left(-\left(\left|x\right| - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(\frac{-1}{x}\right)}, x\right) \]
   8. Step-by-step derivation

      1. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{-1}{x}\right)\right), x\right) \]

      2. /-lowering-/.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(-1, x\right)\right), x\right) \]

   9. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\frac{-1}{x}\right)}, x\right) \]

   if -2 < x < 2

   1. Initial program 24.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 24.1%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 24.1%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 91.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 91.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 15.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Simplified 96.8%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 2 < x

   1. Initial program 53.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log x, x\right) \]

      4. log-lowering-log.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   7. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- x)) x)
   (if (<= x 2.0) (copysign x x) (copysign (log x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 54.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{neg.f32}\left(x\right)\right), x\right) \]

   7. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   if -2 < x < 2

   1. Initial program 24.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 24.1%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 24.1%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 91.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 91.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 15.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Simplified 96.8%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 2 < x

   1. Initial program 53.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log x, x\right) \]

      4. log-lowering-log.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   7. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 2.0) (copysign x x) (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 34.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 49.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 49.7%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 75.8%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   7. Simplified 75.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   9. Applied egg-rr 10.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Simplified 68.1%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 2 < x

   1. Initial program 53.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log x, x\right) \]

      4. log-lowering-log.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   7. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 52.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign x x))

C

float code(float x) {
	return copysignf(x, x);
}

Julia

function code(x)
	return copysign(x, x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(x);
end

Derivation

1. Initial program 38.3%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Step-by-step derivation

   1. copysign-lowering-copysign.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

   2. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

   4. fabs-lowering-fabs.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

   6. hypot-1-def N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   7. hypot-lowering-hypot.f32 59.9%

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

3. Simplified 59.9%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
6. Step-by-step derivation

   1. log1p-define N/A

      \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

   2. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

   3. fabs-lowering-fabs.f32 69.0%

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

7. Simplified 69.0%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

9. Applied egg-rr 9.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot x}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}}}{1 - x \cdot x}\right)}, x\right) \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
11. Step-by-step derivation

12. Simplified 56.2%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
13. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

float code(float x) {
	float t_0 = 1.0f / fabsf(x);
	return copysignf(log1pf((fabsf(x) + (fabsf(x) / (hypotf(1.0f, t_0) + t_0)))), x);
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) / abs(x))
	return copysign(log1p(Float32(abs(x) + Float32(abs(x) / Float32(hypot(Float32(1.0), t_0) + t_0)))), x)
end

Reproduce

herbie shell --seed 2024140 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (! :herbie-platform default (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))