Rust f32::asinh

Percentage Accurate: 38.1% → 99.0%

Time: 14.1s

Alternatives: 14

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: 38.1% 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: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{0.5}{x}\right) + x \cdot \left(1 + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (- (+ (fabs x) (/ (+ -0.5 (/ 0.125 (* x x))) x)) x)) x)
     (if (<= t_0 0.5)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (* x x)
           (+
            -0.16666666666666666
            (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
        x)
       (copysign
        (log
         (+
          (+ (fabs x) (/ 0.5 x))
          (* x (+ 1.0 (/ -0.125 (* x (* x (* x x))))))))
        x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(logf(((fabsf(x) + ((-0.5f + (0.125f / (x * x))) / x)) - x)), x);
	} else if (t_0 <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(logf(((fabsf(x) + (0.5f / x)) + (x * (1.0f + (-0.125f / (x * (x * (x * x)))))))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(abs(x) + Float32(Float32(Float32(-0.5) + Float32(Float32(0.125) / Float32(x * x))) / x)) - x)), x);
	elseif (t_0 <= Float32(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(abs(x) + Float32(Float32(0.5) / x)) + Float32(x * Float32(Float32(1.0) + Float32(Float32(-0.125) / Float32(x * Float32(x * Float32(x * x)))))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = sign(x) * abs(log(((abs(x) + ((single(-0.5) + (single(0.125) / (x * x))) / x)) - x)));
	elseif (t_0 <= single(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(log(((abs(x) + (single(0.5) / x)) + (x * (single(1.0) + (single(-0.125) / (x * (x * (x * x)))))))));
	end
	tmp_2 = 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 62.4%

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

   3. Taylor expanded in x around -inf

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

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

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

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

   5. Simplified 98.7%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.5

   1. Initial program 22.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.3%

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

   5. Simplified 22.3%

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

   7. Applied egg-rr 16.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.6%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.6%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \frac{\left|x\right| + \frac{0.5}{x}}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (- (+ (fabs x) (/ (+ -0.5 (/ 0.125 (* x x))) x)) x)) x)
   (if (<= x 0.5)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (* x (+ 1.0 (/ (+ (fabs x) (/ 0.5 x)) x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf(((fabsf(x) + ((-0.5f + (0.125f / (x * x))) / x)) - x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(logf((x * (1.0f + ((fabsf(x) + (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) + Float32(Float32(Float32(-0.5) + Float32(Float32(0.125) / Float32(x * x))) / x)) - x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(1.0) + Float32(Float32(abs(x) + Float32(Float32(0.5) / 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) + ((single(-0.5) + (single(0.125) / (x * x))) / x)) - x)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(log((x * (single(1.0) + ((abs(x) + (single(0.5) / x)) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 62.4%

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

   3. Taylor expanded in x around -inf

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

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

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

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

   5. Simplified 98.7%

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

   if -1 < x < 0.5

   1. Initial program 22.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.3%

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

   5. Simplified 22.3%

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

   7. Applied egg-rr 16.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.6%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lft-mult-inverse N/A

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

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

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-frac-neg2 N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

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

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

   8. Simplified 99.4%

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

Alternative 3: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \frac{\left|x\right| + \frac{0.5}{x}}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
   (if (<= x 0.5)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (* x (+ 1.0 (/ (+ (fabs x) (/ 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 <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(logf((x * (1.0f + ((fabsf(x) + (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(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(1.0) + Float32(Float32(abs(x) + Float32(Float32(0.5) / 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(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(log((x * (single(1.0) + ((abs(x) + (single(0.5) / x)) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 62.4%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.2%

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

   if -1 < x < 0.5

   1. Initial program 22.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.3%

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

   5. Simplified 22.3%

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

   7. Applied egg-rr 16.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.6%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lft-mult-inverse N/A

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

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

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-frac-neg2 N/A

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

      13. mul-1-neg N/A

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

      14. remove-double-neg N/A

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

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

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

   8. Simplified 99.4%

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

Alternative 4: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
   (if (<= x 0.5)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (+ (/ 0.5 x) (+ x (fabs 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 <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (x + fabsf(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(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(x + abs(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(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (x + abs(x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 62.4%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.2%

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

   if -1 < x < 0.5

   1. Initial program 22.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.3%

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

   5. Simplified 22.3%

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

   7. Applied egg-rr 16.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.6%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lft-mult-inverse N/A

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

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

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

   5. Simplified 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 0.5)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (+ (/ 0.5 x) (+ x (fabs x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (x + 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);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(x + abs(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(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (x + abs(x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 62.4%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   4. 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. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| - x\right)\right), x\right) \]

      15. --lowering--.f32 N/A

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

      16. fabs-lowering-fabs.f32 96.6%

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

   5. Simplified 96.6%

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

   if -1 < x < 0.5

   1. Initial program 22.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.3%

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

   5. Simplified 22.3%

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

   7. Applied egg-rr 16.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.6%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lft-mult-inverse N/A

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

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

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

   5. Simplified 99.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 0.5)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (+ (/ 0.5 x) (/ -0.125 (* x (* x x))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (-0.125f / (x * (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(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(Float32(-0.125) / Float32(x * Float32(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(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (single(-0.125) / (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 62.4%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   4. 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. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| - x\right)\right), x\right) \]

      15. --lowering--.f32 N/A

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

      16. fabs-lowering-fabs.f32 96.6%

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

   5. Simplified 96.6%

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

   if -1 < x < 0.5

   1. Initial program 22.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.3%

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

   5. Simplified 22.3%

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

   7. Applied egg-rr 16.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.6%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      19. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{3}}\right)\right)\right), x\right) \]

      20. metadata-eval N/A

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

      21. /-lowering-/.f32 N/A

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

      22. cube-mult N/A

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

      23. unpow2 N/A

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

   8. Simplified 99.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 37.3%

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

   3. Taylor expanded in x around 0

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

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 74.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 74.9%

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

   if 0.5 < x

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      19. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{3}}\right)\right)\right), x\right) \]

      20. metadata-eval N/A

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

      21. /-lowering-/.f32 N/A

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

      22. cube-mult N/A

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

      23. unpow2 N/A

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

   8. Simplified 99.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

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

Alternative 8: 85.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.600000023841858:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.600000023841858)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 0.5)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (+ (/ 0.5 x) (/ -0.125 (* x (* x x))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.600000023841858f) {
		tmp = copysignf(-logf((-1.0f / x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (-0.125f / (x * (x * x))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.600000023841858))
		tmp = copysign(Float32(-log(Float32(Float32(-1.0) / x))), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(Float32(-0.125) / Float32(x * Float32(x * x))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.600000023841858))
		tmp = sign(x) * abs(-log((single(-1.0) / x)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (single(-0.125) / (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.60000002

   1. Initial program 61.9%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)}, x\right) \]
   4. 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. neg-lowering-neg.f32 N/A

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

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

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

      4. /-lowering-/.f32 43.1%

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

   5. Simplified 43.1%

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

   if -1.60000002 < x < 0.5

   1. Initial program 23.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.4%

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

   5. Simplified 22.4%

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

   7. Applied egg-rr 16.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.0%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      19. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{3}}\right)\right)\right), x\right) \]

      20. metadata-eval N/A

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

      21. /-lowering-/.f32 N/A

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

      22. cube-mult N/A

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

      23. unpow2 N/A

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

   8. Simplified 99.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.600000023841858:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.600000023841858:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{x}{0.5}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.600000023841858)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 0.5)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (- (log (/ x 0.5))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.600000023841858f) {
		tmp = copysignf(-logf((-1.0f / x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(-logf((x / 0.5f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.600000023841858))
		tmp = copysign(Float32(-log(Float32(Float32(-1.0) / x))), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(Float32(-log(Float32(x / Float32(0.5)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.600000023841858))
		tmp = sign(x) * abs(-log((single(-1.0) / x)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(-log((x / single(0.5))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.60000002

   1. Initial program 61.9%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)}, x\right) \]
   4. 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. neg-lowering-neg.f32 N/A

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

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

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

      4. /-lowering-/.f32 43.1%

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

   5. Simplified 43.1%

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

   if -1.60000002 < x < 0.5

   1. Initial program 23.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.4%

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

   5. Simplified 22.4%

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

   7. Applied egg-rr 16.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.0%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lft-mult-inverse N/A

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

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

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 98.4%

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

   8. Simplified 98.4%

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

      1. clear-num N/A

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

      2. log-rec N/A

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

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

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

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

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

      5. /-lowering-/.f32 98.5%

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

   10. Applied egg-rr 98.5%

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

Alternative 10: 85.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.600000023841858:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.600000023841858)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 0.5)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.600000023841858f) {
		tmp = copysignf(-logf((-1.0f / x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.600000023841858))
		tmp = copysign(Float32(-log(Float32(Float32(-1.0) / x))), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.600000023841858))
		tmp = sign(x) * abs(-log((single(-1.0) / x)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.60000002

   1. Initial program 61.9%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)}, x\right) \]
   4. 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. neg-lowering-neg.f32 N/A

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

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

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

      4. /-lowering-/.f32 43.1%

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

   5. Simplified 43.1%

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

   if -1.60000002 < x < 0.5

   1. Initial program 23.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.4%

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

   5. Simplified 22.4%

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

   7. Applied egg-rr 16.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.0%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lft-mult-inverse N/A

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

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

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 98.4%

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

   8. Simplified 98.4%

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

Alternative 11: 85.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.600000023841858:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.600000023841858)
   (copysign (log (- x)) x)
   (if (<= x 0.5)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.600000023841858f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.600000023841858))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.600000023841858))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.60000002

   1. Initial program 61.9%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   4. 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 43.1%

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

   5. Simplified 43.1%

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

   if -1.60000002 < x < 0.5

   1. Initial program 23.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.4%

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

   5. Simplified 22.4%

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

   7. Applied egg-rr 16.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.0%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lft-mult-inverse N/A

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

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

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 98.4%

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

   8. Simplified 98.4%

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

Alternative 12: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.600000023841858:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.600000023841858)
   (copysign (log (- x)) x)
   (if (<= x 0.5)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.600000023841858f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + (x * (x * (0.075f + ((x * x) * -0.044642857142857144f)))))))), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.600000023841858))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(x * Float32(x * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144))))))))), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.600000023841858))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * (single(-0.16666666666666666) + (x * (x * (single(0.075) + ((x * x) * single(-0.044642857142857144))))))))));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.60000002

   1. Initial program 61.9%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   4. 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 43.1%

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

   5. Simplified 43.1%

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

   if -1.60000002 < x < 0.5

   1. Initial program 23.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f32 22.4%

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

   5. Simplified 22.4%

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

   7. Applied egg-rr 16.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      14. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 99.0%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{3}{40}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 99.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   4. 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 44.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   5. Simplified 44.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.5) (copysign x x) (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.5))
		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(0.5))
		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 < 0.5

   1. Initial program 37.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 + \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right| + 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      5. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      9. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left({x}^{2}\right), \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{16} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{16} \cdot {x}^{2} + \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{8} + \frac{1}{16} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      16. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{8}, \left({x}^{2} \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      20. *-lowering-*.f32 18.3%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 18.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + 1\right) + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)}, x\right) \]

   7. Applied egg-rr 12.7%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}} + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right), x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 66.2%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.5 < x

   1. Initial program 47.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   4. 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 44.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   5. Simplified 44.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 54.5% 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 40.0%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)}\right), x\right) \]
4. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right), x\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 + \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right| + 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

   5. fabs-lowering-fabs.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left({x}^{2}\right), \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{16} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{16} \cdot {x}^{2} + \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{-1}{8} + \frac{1}{16} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

   16. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{8}, \left({x}^{2} \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

   18. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

   20. *-lowering-*.f32 16.0%

       \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

5. Simplified 16.0%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + 1\right) + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)}, x\right) \]

7. Applied egg-rr 10.6%

   \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\frac{1 + x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + x \cdot \left(x \cdot x\right)\right)}}} + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right), x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
9. Step-by-step derivation

10. Simplified 51.9%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
11. Add Preprocessing

Developer Target 1: 99.5% 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 2024191 
(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))