Rust f32::asinh

Percentage Accurate: 38.5% → 99.5%

Time: 9.7s

Alternatives: 13

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: 38.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + \mathsf{hypot}\left(1, x\right)\\ t_1 := \log \left({t\_0}^{-0.5}\right)\\ t_2 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_2 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\left(-t\_1\right) - t\_1, x\right)\\ \mathbf{elif}\;t\_2 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log t\_0, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ (fabs x) (hypot 1.0 x)))
        (t_1 (log (pow t_0 -0.5)))
        (t_2 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_2 -0.20000000298023224)
     (copysign (- (- t_1) t_1) x)
     (if (<= t_2 0.10000000149011612)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (* x x)
           (+
            -0.16666666666666666
            (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))))))
        x)
       (copysign (log t_0) x)))))

C

float code(float x) {
	float t_0 = fabsf(x) + hypotf(1.0f, x);
	float t_1 = logf(powf(t_0, -0.5f));
	float t_2 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_2 <= -0.20000000298023224f) {
		tmp = copysignf((-t_1 - t_1), x);
	} else if (t_2 <= 0.10000000149011612f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + ((x * x) * (0.075f + ((x * x) * -0.044642857142857144f))))))), x);
	} else {
		tmp = copysignf(logf(t_0), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(abs(x) + hypot(Float32(1.0), x))
	t_1 = log((t_0 ^ Float32(-0.5)))
	t_2 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_2 <= Float32(-0.20000000298023224))
		tmp = copysign(Float32(Float32(-t_1) - t_1), x);
	elseif (t_2 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(Float32(x * x) * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(t_0), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) + hypot(single(1.0), x);
	t_1 = log((t_0 ^ single(-0.5)));
	t_2 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_2 <= single(-0.20000000298023224))
		tmp = sign(x) * abs((-t_1 - t_1));
	elseif (t_2 <= single(0.10000000149011612))
		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(t_0));
	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) < -0.200000003

   1. Initial program 56.2%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

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

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

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

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

   4. Applied egg-rr 98.3%

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

      1. inv-pow N/A

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

      2. sqr-pow N/A

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

      3. log-prod N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

      10. hypot-undefine N/A

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

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

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

   6. Applied egg-rr 98.4%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.100000001

   1. Initial program 20.6%

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

   3. Taylor expanded in x around 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 20.5%

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

   5. Simplified 20.5%

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

   7. Applied egg-rr 14.7%

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

   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. *-lowering-*.f32 N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 100.0%

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

   10. Simplified 100.0%

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

   if 0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 42.3%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\left(-\log \left({\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{-0.5}\right)\right) - \log \left({\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{-0.5}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))
   (if (<= t_0 -0.20000000298023224)
     t_1
     (if (<= t_0 0.10000000149011612)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (* x x)
           (+
            -0.16666666666666666
            (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))))))
        x)
       t_1))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = t_1;
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + ((x * x) * (0.075f + ((x * x) * -0.044642857142857144f))))))), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = t_1;
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(Float32(x * x) * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144)))))))), x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	t_1 = sign(x) * abs(log((abs(x) + hypot(single(1.0), x))));
	tmp = single(0.0);
	if (t_0 <= single(-0.20000000298023224))
		tmp = t_1;
	elseif (t_0 <= single(0.10000000149011612))
		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 = t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -0.200000003 or 0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 49.0%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 99.1%

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

   3. Simplified 99.1%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.100000001

   1. Initial program 20.6%

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

   3. Taylor expanded in x around 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 20.5%

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

   5. Simplified 20.5%

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

   7. Applied egg-rr 14.7%

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

   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. *-lowering-*.f32 N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}}{x} + -1\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign
    (log (* x (+ (/ (- (fabs x) (/ (+ 0.5 (/ -0.125 (* x x))) x)) x) -1.0)))
    x)
   (if (<= x 1.0)
     (copysign
      (log1p
       (+
        (fabs x)
        (* (* x x) (+ 0.5 (* (* x x) (+ -0.125 (* (* x x) 0.0625)))))))
      x)
     (copysign
      (log
       (+
        (fabs x)
        (* x (+ 1.0 (+ (/ 0.5 (* x x)) (/ -0.125 (* x (* x (* x x)))))))))
      x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((x * (((fabsf(x) - ((0.5f + (-0.125f / (x * x))) / x)) / x) + -1.0f))), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(log1pf((fabsf(x) + ((x * x) * (0.5f + ((x * x) * (-0.125f + ((x * x) * 0.0625f))))))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x * (1.0f + ((0.5f / (x * x)) + (-0.125f / (x * (x * (x * x))))))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(x * Float32(Float32(Float32(abs(x) - Float32(Float32(Float32(0.5) + Float32(Float32(-0.125) / Float32(x * x))) / x)) / x) + Float32(-1.0)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(Float32(x * x) * Float32(Float32(-0.125) + Float32(Float32(x * x) * Float32(0.0625)))))))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x * Float32(Float32(1.0) + Float32(Float32(Float32(0.5) / Float32(x * x)) + Float32(Float32(-0.125) / Float32(x * Float32(x * Float32(x * x))))))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 55.6%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

   5. Simplified 98.2%

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

   if -5 < x < 1

   1. Initial program 23.4%

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

   3. Taylor expanded in x around 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 21.9%

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

   5. Simplified 21.9%

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

   6. Taylor expanded in x around 0

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

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

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

   8. Simplified 98.4%

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

   if 1 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. metadata-eval N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. unpow2 N/A

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

      19. cube-mult N/A

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

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

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

      21. cube-mult N/A

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

      22. unpow2 N/A

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

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

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

      24. unpow2 N/A

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

      25. *-lowering-*.f32 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}}{x} + -1\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}}{x} + -1\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign
    (log (* x (+ (/ (- (fabs x) (/ (+ 0.5 (/ -0.125 (* x x))) x)) x) -1.0)))
    x)
   (if (<= x 1.0)
     (copysign
      (log1p
       (+
        (fabs x)
        (* (* x x) (+ 0.5 (* (* x x) (+ -0.125 (* (* x x) 0.0625)))))))
      x)
     (copysign (log (* x (+ (/ (fabs x) x) (+ 1.0 (/ 0.5 (* x x)))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((x * (((fabsf(x) - ((0.5f + (-0.125f / (x * x))) / x)) / x) + -1.0f))), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(log1pf((fabsf(x) + ((x * x) * (0.5f + ((x * x) * (-0.125f + ((x * x) * 0.0625f))))))), x);
	} else {
		tmp = copysignf(logf((x * ((fabsf(x) / x) + (1.0f + (0.5f / (x * x)))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(x * Float32(Float32(Float32(abs(x) - Float32(Float32(Float32(0.5) + Float32(Float32(-0.125) / Float32(x * x))) / x)) / x) + Float32(-1.0)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(Float32(x * x) * Float32(Float32(-0.125) + Float32(Float32(x * x) * Float32(0.0625)))))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(abs(x) / x) + Float32(Float32(1.0) + Float32(Float32(0.5) / Float32(x * x)))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 55.6%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

   5. Simplified 98.2%

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

   if -5 < x < 1

   1. Initial program 23.4%

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

   3. Taylor expanded in x around 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 21.9%

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

   5. Simplified 21.9%

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

   6. Taylor expanded in x around 0

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

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

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

   8. Simplified 98.4%

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

   if 1 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around 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. *-lowering-*.f32 N/A

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f32 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}}{x} + -1\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\left|x\right|}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign
    (log (* x (+ (/ (- (fabs x) (/ (+ 0.5 (/ -0.125 (* x x))) x)) x) -1.0)))
    x)
   (if (<= x 1.0)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))))))
      x)
     (copysign (log (* x (+ (/ (fabs x) x) (+ 1.0 (/ 0.5 (* x x)))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((x * (((fabsf(x) - ((0.5f + (-0.125f / (x * x))) / x)) / x) + -1.0f))), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + ((x * x) * (0.075f + ((x * x) * -0.044642857142857144f))))))), x);
	} else {
		tmp = copysignf(logf((x * ((fabsf(x) / x) + (1.0f + (0.5f / (x * x)))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(x * Float32(Float32(Float32(abs(x) - Float32(Float32(Float32(0.5) + Float32(Float32(-0.125) / Float32(x * x))) / x)) / x) + Float32(-1.0)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(Float32(x * x) * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(abs(x) / x) + Float32(Float32(1.0) + Float32(Float32(0.5) / Float32(x * x)))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((x * (((abs(x) - ((single(0.5) + (single(-0.125) / (x * x))) / x)) / x) + single(-1.0)))));
	elseif (x <= single(1.0))
		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 * ((abs(x) / x) + (single(1.0) + (single(0.5) / (x * x)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 55.6%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

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

   5. Simplified 98.2%

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

   if -5 < x < 1

   1. Initial program 23.4%

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

   3. Taylor expanded in x around 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 21.9%

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

   5. Simplified 21.9%

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

   7. Applied egg-rr 16.3%

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

   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. *-lowering-*.f32 N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 98.4%

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

   10. Simplified 98.4%

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

   if 1 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around 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. *-lowering-*.f32 N/A

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f32 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 98.5%

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

Alternative 6: 98.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))))))
      x)
     (copysign (log (* x (+ (/ (fabs x) x) (+ 1.0 (/ 0.5 (* x x)))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + ((x * x) * (0.075f + ((x * x) * -0.044642857142857144f))))))), x);
	} else {
		tmp = copysignf(logf((x * ((fabsf(x) / x) + (1.0f + (0.5f / (x * x)))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(Float32(x * x) * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(abs(x) / x) + Float32(Float32(1.0) + Float32(Float32(0.5) / Float32(x * x)))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		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 * ((abs(x) / x) + (single(1.0) + (single(0.5) / (x * x)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 55.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   5. Simplified 55.1%

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

   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. metadata-eval N/A

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

      2. *-inverses N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. sqr-abs N/A

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

      9. unpow2 N/A

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

      10. *-inverses N/A

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

      11. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

   if -5 < x < 1

   1. Initial program 23.4%

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

   3. Taylor expanded in x around 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 21.9%

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

   5. Simplified 21.9%

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

   7. Applied egg-rr 16.3%

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

   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. *-lowering-*.f32 N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 98.4%

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

   10. Simplified 98.4%

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

   if 1 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around 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. *-lowering-*.f32 N/A

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f32 99.0%

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

   5. Simplified 99.0%

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

Alternative 7: 98.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))))))
      x)
     (copysign (log (+ (fabs x) (* x (+ 1.0 (/ 0.5 (* x x)))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + ((x * x) * (0.075f + ((x * x) * -0.044642857142857144f))))))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x * (1.0f + (0.5f / (x * x)))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(Float32(x * x) * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x * Float32(Float32(1.0) + Float32(Float32(0.5) / Float32(x * x)))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		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) + (x * (single(1.0) + (single(0.5) / (x * x)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 55.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   5. Simplified 55.1%

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

   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. metadata-eval N/A

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

      2. *-inverses N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. sqr-abs N/A

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

      9. unpow2 N/A

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

      10. *-inverses N/A

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

      11. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

   if -5 < x < 1

   1. Initial program 23.4%

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

   3. Taylor expanded in x around 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 21.9%

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

   5. Simplified 21.9%

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

   7. Applied egg-rr 16.3%

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

   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. *-lowering-*.f32 N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 98.4%

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

   10. Simplified 98.4%

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

   if 1 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 99.0%

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

   5. Simplified 99.0%

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

Alternative 8: 98.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))))))
      x)
     (copysign (log (+ x (fabs x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * (-0.16666666666666666f + ((x * x) * (0.075f + ((x * x) * -0.044642857142857144f))))))), x);
	} else {
		tmp = copysignf(logf((x + fabsf(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(Float32(x * x) * Float32(Float32(0.075) + Float32(Float32(x * x) * Float32(-0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(Float32(x + abs(x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		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 + abs(x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 55.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   5. Simplified 55.1%

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

   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. metadata-eval N/A

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

      2. *-inverses N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. sqr-abs N/A

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

      9. unpow2 N/A

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

      10. *-inverses N/A

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

      11. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

   if -5 < x < 1

   1. Initial program 23.4%

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

   3. Taylor expanded in x around 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 21.9%

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

   5. Simplified 21.9%

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

   7. Applied egg-rr 16.3%

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

   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. *-lowering-*.f32 N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 98.4%

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

   10. Simplified 98.4%

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

   if 1 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.4%

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

4. Final simplification 98.1%

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

Alternative 9: 98.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (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 <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		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(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(Float32(x * 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(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		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 < -5

   1. Initial program 55.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   5. Simplified 55.1%

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

   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. metadata-eval N/A

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

      2. *-inverses N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. sqr-abs N/A

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

      9. unpow2 N/A

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

      10. *-inverses N/A

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

      11. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

   if -5 < x < 1

   1. Initial program 23.4%

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

   3. Taylor expanded in x around 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 21.9%

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

   5. Simplified 21.9%

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

   7. Applied egg-rr 16.3%

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

   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. *-lowering-*.f32 N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 98.4%

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

   10. Simplified 98.4%

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

   if 1 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around 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. *-lowering-*.f32 N/A

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f32 99.0%

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

   5. Simplified 99.0%

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

   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 97.4%

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

   8. Simplified 97.4%

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

Alternative 10: 84.9% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (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 <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		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(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(Float32(x * 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(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		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 < -5

   1. Initial program 55.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   5. Simplified 55.1%

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

   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. metadata-eval N/A

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

      2. *-inverses N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. sqr-abs N/A

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

      9. unpow2 N/A

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

      10. *-inverses N/A

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

      11. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

   if -5 < x < 1

   1. Initial program 23.4%

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

   3. Taylor expanded in x around 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 21.9%

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

   5. Simplified 21.9%

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

   7. Applied egg-rr 16.3%

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

   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. *-lowering-*.f32 N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 98.4%

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

   10. Simplified 98.4%

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

   if 1 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around 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 44.9%

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

   5. Simplified 44.9%

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

Alternative 11: 71.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (- x)) x)
   (if (<= x 1.0)
     (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 <= -5.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 1.0f) {
		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(-5.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) + Float32(Float32(x * 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(-5.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(1.0))
		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 < -5

   1. Initial program 55.6%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   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 43.8%

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

   5. Simplified 43.8%

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

   if -5 < x < 1

   1. Initial program 23.4%

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

   3. Taylor expanded in x around 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 21.9%

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

   5. Simplified 21.9%

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

   7. Applied egg-rr 16.3%

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

   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. *-lowering-*.f32 N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 98.4%

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

   10. Simplified 98.4%

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

   if 1 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around 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 44.9%

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

   5. Simplified 44.9%

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

Alternative 12: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 5.0) (copysign x x) (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 5.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(5.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(5.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5

   1. Initial program 33.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 18.6%

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

   5. Simplified 18.6%

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

   7. Applied egg-rr 13.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 70.9%

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

   if 5 < x

   1. Initial program 37.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 45.2%

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

   5. Simplified 45.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 53.7% 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 34.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 16.2%

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

5. Simplified 16.2%

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

7. Applied egg-rr 10.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
9. Step-by-step derivation

10. Simplified 56.7%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
11. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

float code(float x) {
	float t_0 = 1.0f / fabsf(x);
	return copysignf(log1pf((fabsf(x) + (fabsf(x) / (hypotf(1.0f, t_0) + t_0)))), x);
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) / abs(x))
	return copysign(log1p(Float32(abs(x) + Float32(abs(x) / Float32(hypot(Float32(1.0), t_0) + t_0)))), x)
end

Reproduce

herbie shell --seed 2024154 
(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))