Rust f32::asinh

Percentage Accurate: 38.1% → 99.3%

Time: 6.5s

Alternatives: 12

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 38.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 - {x}^{2} \cdot 0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|{\left(\sqrt[3]{x}\right)}^{2}\right|, \left|\sqrt[3]{x}\right|, \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (pow x 2.0)
           (-
            (* (pow x 2.0) (- 0.075 (* (pow x 2.0) 0.044642857142857144)))
            0.16666666666666666))))
        x)
       (copysign
        (log (fma (fabs (pow (cbrt x) 2.0)) (fabs (cbrt x)) (hypot 1.0 x)))
        x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((powf(x, 2.0f) * (0.075f - (powf(x, 2.0f) * 0.044642857142857144f))) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf(fmaf(fabsf(powf(cbrtf(x), 2.0f)), fabsf(cbrtf(x)), hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(Float32(0.075) - Float32((x ^ Float32(2.0)) * Float32(0.044642857142857144)))) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(log(fma(abs((cbrt(x) ^ Float32(2.0))), abs(cbrt(x)), hypot(Float32(1.0), x))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. hypot-1-def 98.7%

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

   3. Simplified 98.7%

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

      1. flip-+ 9.5%

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

      2. clear-num 9.5%

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

      3. log-div 9.5%

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

      4. metadata-eval 9.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr -0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 10.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 10.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 10.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 10.9%

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

      4. unpow2 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 10.9%

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

      13. unpow2 10.9%

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

      14. associate--r+ 49.3%

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

      15. +-inverses 98.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 98.7%

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

      17. *-rgt-identity 98.7%

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

      18. associate-/l* 98.7%

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

      19. metadata-eval 98.7%

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

      20. *-commutative 98.7%

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

      21. neg-mul-1 98.7%

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

   8. Simplified 98.7%

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

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

   1. Initial program 18.5%

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

      1. +-commutative 18.5%

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

      2. hypot-1-def 18.5%

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

   3. Simplified 18.5%

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

      1. flip-+ 18.7%

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

      2. clear-num 18.7%

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

      3. log-div 19.2%

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

      4. metadata-eval 19.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr 9.2%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 19.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 19.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr 9.2%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 19.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 19.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 19.1%

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

      4. unpow2 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 19.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 19.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 19.1%

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

      13. unpow2 19.1%

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

      14. associate--r+ 19.1%

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

      15. +-inverses 19.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 19.1%

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

      17. *-rgt-identity 19.1%

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

      18. associate-/l* 19.1%

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

      19. metadata-eval 19.1%

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

      20. *-commutative 19.1%

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

      21. neg-mul-1 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{x \cdot \left({x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.044642857142857144 \cdot {x}^{2} - 0.075\right)\right) - 1\right)}, x\right) \]

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

   1. Initial program 58.9%

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

      1. +-commutative 58.9%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      2. fabs-mul 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left|\sqrt[3]{x} \cdot \sqrt[3]{x}\right| \cdot \left|\sqrt[3]{x}\right|} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      3. fma-define 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|\sqrt[3]{x} \cdot \sqrt[3]{x}\right|, \left|\sqrt[3]{x}\right|, \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      4. pow2 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}\right|, \left|\sqrt[3]{x}\right|, \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|{\left(\sqrt[3]{x}\right)}^{2}\right|, \left|\sqrt[3]{x}\right|, \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 - {x}^{2} \cdot 0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|{\left(\sqrt[3]{x}\right)}^{2}\right|, \left|\sqrt[3]{x}\right|, \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 - {x}^{2} \cdot 0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (pow x 2.0)
           (-
            (* (pow x 2.0) (- 0.075 (* (pow x 2.0) 0.044642857142857144)))
            0.16666666666666666))))
        x)
       (copysign (log (pow (cbrt (+ x (hypot 1.0 x))) 3.0)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((powf(x, 2.0f) * (0.075f - (powf(x, 2.0f) * 0.044642857142857144f))) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf(powf(cbrtf((x + hypotf(1.0f, x))), 3.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(Float32(0.075) - Float32((x ^ Float32(2.0)) * Float32(0.044642857142857144)))) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(log((cbrt(Float32(x + hypot(Float32(1.0), x))) ^ Float32(3.0))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. hypot-1-def 98.7%

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

   3. Simplified 98.7%

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

      1. flip-+ 9.5%

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

      2. clear-num 9.5%

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

      3. log-div 9.5%

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

      4. metadata-eval 9.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr -0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 10.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 10.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 10.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 10.9%

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

      4. unpow2 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 10.9%

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

      13. unpow2 10.9%

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

      14. associate--r+ 49.3%

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

      15. +-inverses 98.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 98.7%

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

      17. *-rgt-identity 98.7%

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

      18. associate-/l* 98.7%

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

      19. metadata-eval 98.7%

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

      20. *-commutative 98.7%

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

      21. neg-mul-1 98.7%

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

   8. Simplified 98.7%

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

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

   1. Initial program 18.5%

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

      1. +-commutative 18.5%

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

      2. hypot-1-def 18.5%

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

   3. Simplified 18.5%

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

      1. flip-+ 18.7%

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

      2. clear-num 18.7%

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

      3. log-div 19.2%

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

      4. metadata-eval 19.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr 9.2%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 19.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 19.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr 9.2%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 19.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 19.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 19.1%

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

      4. unpow2 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 19.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 19.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 19.1%

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

      13. unpow2 19.1%

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

      14. associate--r+ 19.1%

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

      15. +-inverses 19.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 19.1%

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

      17. *-rgt-identity 19.1%

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

      18. associate-/l* 19.1%

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

      19. metadata-eval 19.1%

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

      20. *-commutative 19.1%

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

      21. neg-mul-1 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{x \cdot \left({x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.044642857142857144 \cdot {x}^{2} - 0.075\right)\right) - 1\right)}, x\right) \]

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

   1. Initial program 58.9%

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

      1. +-commutative 58.9%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 99.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt[3]{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

      2. pow3 99.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

      3. add-sqr-sqrt 99.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right), x\right) \]

      4. fabs-sqr 99.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 99.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right), x\right) \]

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 - {x}^{2} \cdot 0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 - {x}^{2} \cdot 0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (pow x 2.0)
           (-
            (* (pow x 2.0) (- 0.075 (* (pow x 2.0) 0.044642857142857144)))
            0.16666666666666666))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((powf(x, 2.0f) * (0.075f - (powf(x, 2.0f) * 0.044642857142857144f))) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(Float32(0.075) - Float32((x ^ Float32(2.0)) * Float32(0.044642857142857144)))) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x ^ single(2.0)) * (single(0.075) - ((x ^ single(2.0)) * single(0.044642857142857144)))) - single(0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. hypot-1-def 98.7%

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

   3. Simplified 98.7%

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

      1. flip-+ 9.5%

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

      2. clear-num 9.5%

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

      3. log-div 9.5%

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

      4. metadata-eval 9.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr -0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 10.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 10.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 10.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 10.9%

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

      4. unpow2 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 10.9%

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

      13. unpow2 10.9%

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

      14. associate--r+ 49.3%

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

      15. +-inverses 98.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 98.7%

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

      17. *-rgt-identity 98.7%

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

      18. associate-/l* 98.7%

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

      19. metadata-eval 98.7%

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

      20. *-commutative 98.7%

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

      21. neg-mul-1 98.7%

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

   8. Simplified 98.7%

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

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

   1. Initial program 18.5%

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

      1. +-commutative 18.5%

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

      2. hypot-1-def 18.5%

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

   3. Simplified 18.5%

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

      1. flip-+ 18.7%

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

      2. clear-num 18.7%

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

      3. log-div 19.2%

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

      4. metadata-eval 19.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr 9.2%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 19.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 19.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr 9.2%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 19.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 19.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 19.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 19.1%

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

      4. unpow2 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 19.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 19.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 19.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 19.1%

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

      13. unpow2 19.1%

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

      14. associate--r+ 19.1%

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

      15. +-inverses 19.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 19.1%

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

      17. *-rgt-identity 19.1%

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

      18. associate-/l* 19.1%

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

      19. metadata-eval 19.1%

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

      20. *-commutative 19.1%

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

      21. neg-mul-1 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{x \cdot \left({x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.044642857142857144 \cdot {x}^{2} - 0.075\right)\right) - 1\right)}, x\right) \]

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

   1. Initial program 58.9%

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

      1. +-commutative 58.9%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-sqr-sqrt 99.8%

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

      5. fabs-sqr 99.8%

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

      6. add-sqr-sqrt 99.9%

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

      7. metadata-eval 99.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 - {x}^{2} \cdot 0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.004999999888241291)
       (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.004999999888241291f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.004999999888241291))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.004999999888241291))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. hypot-1-def 98.7%

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

   3. Simplified 98.7%

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

      1. flip-+ 9.5%

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

      2. clear-num 9.5%

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

      3. log-div 9.5%

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

      4. metadata-eval 9.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr -0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 10.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 10.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 10.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 10.9%

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

      4. unpow2 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 10.9%

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

      13. unpow2 10.9%

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

      14. associate--r+ 49.3%

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

      15. +-inverses 98.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 98.7%

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

      17. *-rgt-identity 98.7%

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

      18. associate-/l* 98.7%

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

      19. metadata-eval 98.7%

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

      20. *-commutative 98.7%

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

      21. neg-mul-1 98.7%

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

   8. Simplified 98.7%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.00499999989

   1. Initial program 17.9%

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

      1. +-commutative 17.9%

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

      2. hypot-1-def 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in x around 0 18.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   6. Step-by-step derivation

      1. +-commutative 18.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)}, x\right) \]

      2. fma-define 18.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. rem-square-sqrt 8.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. fabs-sqr 8.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. rem-square-sqrt 18.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \color{blue}{x}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. log1p-define 99.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      7. rem-square-sqrt 39.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      8. fabs-sqr 39.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      9. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot x + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x}, x\right) \]

      2. *-lft-identity 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x, x\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]

      4. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.16666666666666666}, x\right) \]

      6. unpow2 100.0%

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

      7. cube-mult 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{{x}^{3}} \cdot -0.16666666666666666, x\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x + {x}^{3} \cdot -0.16666666666666666}, x\right) \]

   if 0.00499999989 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 59.5%

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

      1. +-commutative 59.5%

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

      2. hypot-1-def 99.8%

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

   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
   5. Step-by-step derivation

      1. *-un-lft-identity 99.8%

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

      2. *-commutative 99.8%

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

      3. log-prod 99.8%

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

      4. add-sqr-sqrt 99.7%

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

      5. fabs-sqr 99.7%

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

      6. add-sqr-sqrt 99.8%

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

      7. metadata-eval 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 0.004999999888241291)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 0.004999999888241291f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(0.004999999888241291))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(0.004999999888241291))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. hypot-1-def 98.7%

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

   3. Simplified 98.7%

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

      1. flip-+ 9.5%

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

      2. clear-num 9.5%

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

      3. log-div 9.5%

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

      4. metadata-eval 9.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr -0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 10.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 10.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 10.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 10.9%

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

      4. unpow2 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 10.9%

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

      13. unpow2 10.9%

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

      14. associate--r+ 49.3%

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

      15. +-inverses 98.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 98.7%

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

      17. *-rgt-identity 98.7%

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

      18. associate-/l* 98.7%

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

      19. metadata-eval 98.7%

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

      20. *-commutative 98.7%

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

      21. neg-mul-1 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in x around -inf 94.6%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-commutative 94.6%

          \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   11. Simplified 94.6%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   if -2 < x < 0.00499999989

   1. Initial program 17.9%

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

      1. +-commutative 17.9%

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

      2. hypot-1-def 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in x around 0 18.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   6. Step-by-step derivation

      1. +-commutative 18.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)}, x\right) \]

      2. fma-define 18.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. rem-square-sqrt 8.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. fabs-sqr 8.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. rem-square-sqrt 18.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \color{blue}{x}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. log1p-define 99.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      7. rem-square-sqrt 39.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      8. fabs-sqr 39.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      9. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot x + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x}, x\right) \]

      2. *-lft-identity 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x, x\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]

      4. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.16666666666666666}, x\right) \]

      6. unpow2 100.0%

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

      7. cube-mult 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{{x}^{3}} \cdot -0.16666666666666666, x\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x + {x}^{3} \cdot -0.16666666666666666}, x\right) \]

   if 0.00499999989 < x

   1. Initial program 59.5%

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

      1. +-commutative 59.5%

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

      2. hypot-1-def 99.8%

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

   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
   5. Step-by-step derivation

      1. *-un-lft-identity 99.8%

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

      2. *-commutative 99.8%

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

      3. log-prod 99.8%

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

      4. add-sqr-sqrt 99.7%

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

      5. fabs-sqr 99.7%

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

      6. add-sqr-sqrt 99.8%

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

      7. metadata-eval 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \frac{x}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 0.5)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (* x (+ 1.0 (/ x x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x * (1.0f + (x / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(1.0) + Float32(x / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x * (single(1.0) + (x / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. hypot-1-def 98.7%

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

   3. Simplified 98.7%

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

      1. flip-+ 9.5%

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

      2. clear-num 9.5%

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

      3. log-div 9.5%

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

      4. metadata-eval 9.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr -0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 10.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 10.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 10.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 10.9%

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

      4. unpow2 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 10.9%

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

      13. unpow2 10.9%

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

      14. associate--r+ 49.3%

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

      15. +-inverses 98.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 98.7%

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

      17. *-rgt-identity 98.7%

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

      18. associate-/l* 98.7%

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

      19. metadata-eval 98.7%

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

      20. *-commutative 98.7%

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

      21. neg-mul-1 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in x around -inf 94.6%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-commutative 94.6%

          \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   11. Simplified 94.6%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   if -2 < x < 0.5

   1. Initial program 19.0%

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

      1. +-commutative 19.0%

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

      2. hypot-1-def 19.0%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in x around 0 19.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   6. Step-by-step derivation

      1. +-commutative 19.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)}, x\right) \]

      2. fma-define 19.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. rem-square-sqrt 9.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. fabs-sqr 9.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. rem-square-sqrt 19.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \color{blue}{x}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. log1p-define 99.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      7. rem-square-sqrt 39.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      8. fabs-sqr 39.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      9. rem-square-sqrt 99.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   7. Simplified 99.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot x + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x}, x\right) \]

      2. *-lft-identity 99.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x, x\right) \]

      3. *-commutative 99.5%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]

      4. *-commutative 99.5%

         \[\leadsto \mathsf{copysign}\left(x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

      5. associate-*r* 99.5%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.16666666666666666}, x\right) \]

      6. unpow2 99.5%

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

      7. cube-mult 99.5%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{{x}^{3}} \cdot -0.16666666666666666, x\right) \]

   10. Simplified 99.5%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x + {x}^{3} \cdot -0.16666666666666666}, x\right) \]

   if 0.5 < x

   1. Initial program 58.2%

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

      1. +-commutative 58.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. rem-square-sqrt 99.2%

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

      2. fabs-sqr 99.2%

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

      3. rem-square-sqrt 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \frac{x}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 0.5)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (- (log (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(-logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	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 < -2

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. hypot-1-def 98.7%

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

   3. Simplified 98.7%

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

      1. flip-+ 9.5%

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

      2. clear-num 9.5%

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

      3. log-div 9.5%

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

      4. metadata-eval 9.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr -0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 10.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 10.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 10.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 10.9%

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

      4. unpow2 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 10.9%

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

      13. unpow2 10.9%

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

      14. associate--r+ 49.3%

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

      15. +-inverses 98.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 98.7%

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

      17. *-rgt-identity 98.7%

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

      18. associate-/l* 98.7%

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

      19. metadata-eval 98.7%

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

      20. *-commutative 98.7%

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

      21. neg-mul-1 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in x around -inf 94.6%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-commutative 94.6%

          \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   11. Simplified 94.6%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   if -2 < x < 0.5

   1. Initial program 19.0%

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

      1. +-commutative 19.0%

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

      2. hypot-1-def 19.0%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in x around 0 19.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   6. Step-by-step derivation

      1. +-commutative 19.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)}, x\right) \]

      2. fma-define 19.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. rem-square-sqrt 9.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. fabs-sqr 9.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. rem-square-sqrt 19.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \color{blue}{x}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. log1p-define 99.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      7. rem-square-sqrt 39.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      8. fabs-sqr 39.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      9. rem-square-sqrt 99.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   7. Simplified 99.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot x + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x}, x\right) \]

      2. *-lft-identity 99.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x, x\right) \]

      3. *-commutative 99.5%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]

      4. *-commutative 99.5%

         \[\leadsto \mathsf{copysign}\left(x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

      5. associate-*r* 99.5%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.16666666666666666}, x\right) \]

      6. unpow2 99.5%

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

      7. cube-mult 99.5%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{{x}^{3}} \cdot -0.16666666666666666, x\right) \]

   10. Simplified 99.5%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x + {x}^{3} \cdot -0.16666666666666666}, x\right) \]

   if 0.5 < x

   1. Initial program 58.2%

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

      1. +-commutative 58.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 2.5%

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

      2. clear-num 2.5%

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

      3. log-div 2.5%

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

      4. metadata-eval 2.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt 3.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr 3.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 2.5%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 2.5%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt 4.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr 4.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 2.5%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 2.5%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 2.5%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 2.4%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 2.4%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 2.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 2.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 2.4%

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

      4. unpow2 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 2.4%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 2.4%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 2.4%

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

      13. unpow2 2.4%

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

      14. associate--r+ 5.8%

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

      15. +-inverses 8.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 8.7%

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

      17. *-rgt-identity 8.7%

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

      18. associate-/l* 8.7%

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

      19. metadata-eval 8.7%

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

      20. *-commutative 8.7%

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

      21. neg-mul-1 8.7%

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

   8. Simplified 8.7%

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

   9. Taylor expanded in x around inf 99.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 0.5) (copysign x x) (copysign (- (log (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(-logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs(x);
	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 < -2

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. hypot-1-def 98.7%

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

   3. Simplified 98.7%

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

      1. flip-+ 9.5%

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

      2. clear-num 9.5%

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

      3. log-div 9.5%

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

      4. metadata-eval 9.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr -0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 10.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 10.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 10.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 10.9%

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

      4. unpow2 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 10.9%

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

      13. unpow2 10.9%

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

      14. associate--r+ 49.3%

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

      15. +-inverses 98.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 98.7%

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

      17. *-rgt-identity 98.7%

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

      18. associate-/l* 98.7%

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

      19. metadata-eval 98.7%

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

      20. *-commutative 98.7%

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

      21. neg-mul-1 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in x around -inf 94.6%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-commutative 94.6%

          \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   11. Simplified 94.6%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   if -2 < x < 0.5

   1. Initial program 19.0%

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

      1. +-commutative 19.0%

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

      2. hypot-1-def 19.0%

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

   3. Simplified 19.0%

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

      1. flip-+ 19.3%

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

      2. clear-num 19.3%

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

      3. log-div 19.8%

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

      4. metadata-eval 19.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt 9.9%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr 9.9%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 19.7%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 19.7%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt 9.9%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr 9.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 19.7%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 19.7%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 19.7%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 19.7%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 19.7%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 19.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 19.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 19.7%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 19.7%

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

      4. unpow2 19.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 19.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 19.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 19.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 19.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 19.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 19.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 19.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 19.7%

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

      13. unpow2 19.7%

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

      14. associate--r+ 19.7%

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

      15. +-inverses 19.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 19.7%

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

      17. *-rgt-identity 19.7%

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

      18. associate-/l* 19.7%

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

      19. metadata-eval 19.7%

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

      20. *-commutative 19.7%

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

      21. neg-mul-1 19.7%

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

   8. Simplified 19.7%

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

   9. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{-1 \cdot x}, x\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 99.0%

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

   11. Simplified 99.0%

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

   if 0.5 < x

   1. Initial program 58.2%

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

      1. +-commutative 58.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 2.5%

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

      2. clear-num 2.5%

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

      3. log-div 2.5%

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

      4. metadata-eval 2.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt 3.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr 3.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 2.5%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 2.5%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt 4.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr 4.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 2.5%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 2.5%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 2.5%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 2.4%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 2.4%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 2.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 2.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 2.4%

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

      4. unpow2 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 2.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 2.4%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 2.4%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 2.4%

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

      13. unpow2 2.4%

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

      14. associate--r+ 5.8%

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

      15. +-inverses 8.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 8.7%

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

      17. *-rgt-identity 8.7%

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

      18. associate-/l* 8.7%

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

      19. metadata-eval 8.7%

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

      20. *-commutative 8.7%

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

      21. neg-mul-1 8.7%

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

   8. Simplified 8.7%

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

   9. Taylor expanded in x around inf 99.2%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0) (copysign (- (log (* x -2.0))) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -2

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. hypot-1-def 98.7%

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

   3. Simplified 98.7%

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

      1. flip-+ 9.5%

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

      2. clear-num 9.5%

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

      3. log-div 9.5%

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

      4. metadata-eval 9.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 9.8%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr -0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 10.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 9.8%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 10.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 10.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 10.9%

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

      4. unpow2 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 10.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 10.9%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 10.9%

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

      13. unpow2 10.9%

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

      14. associate--r+ 49.3%

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

      15. +-inverses 98.7%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 98.7%

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

      17. *-rgt-identity 98.7%

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

      18. associate-/l* 98.7%

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

      19. metadata-eval 98.7%

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

      20. *-commutative 98.7%

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

      21. neg-mul-1 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in x around -inf 94.6%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-commutative 94.6%

          \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   11. Simplified 94.6%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   if -2 < x

   1. Initial program 31.0%

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

      1. +-commutative 31.0%

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

      2. hypot-1-def 43.8%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in x around 0 26.1%

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

      1. log1p-define 80.6%

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

      2. rem-square-sqrt 40.2%

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

      3. fabs-sqr 40.2%

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

      4. rem-square-sqrt 80.6%

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

   7. Simplified 80.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0) (copysign (log (- x)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(-x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -2

   1. Initial program 52.5%

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

      1. +-commutative 52.5%

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

      2. hypot-1-def 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in x around -inf 43.8%

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

      1. mul-1-neg 43.8%

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

   7. Simplified 43.8%

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

   if -2 < x

   1. Initial program 31.0%

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

      1. +-commutative 31.0%

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

      2. hypot-1-def 43.8%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in x around 0 26.1%

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

      1. log1p-define 80.6%

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

      2. rem-square-sqrt 40.2%

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

      3. fabs-sqr 40.2%

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

      4. rem-square-sqrt 80.6%

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

   7. Simplified 80.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.5) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 31.3%

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

      1. +-commutative 31.3%

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

      2. hypot-1-def 48.1%

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

   3. Simplified 48.1%

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

      1. flip-+ 15.7%

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

      2. clear-num 15.7%

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

      3. log-div 16.0%

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

      4. metadata-eval 16.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. add-sqr-sqrt 6.3%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. fabs-sqr 6.3%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 16.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. pow2 16.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt 6.3%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr 6.3%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 16.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 16.2%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 16.1%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 16.5%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 16.5%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 16.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 16.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      2. div-sub 16.5%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

      3. fma-undefine 16.5%

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

      4. unpow2 16.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      5. associate--r+ 16.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      6. +-inverses 16.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 16.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      8. *-rgt-identity 16.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      9. associate-/l* 16.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      10. metadata-eval 16.5%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. *-commutative 16.5%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      12. fma-undefine 16.5%

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

      13. unpow2 16.5%

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

      14. associate--r+ 30.5%

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

      15. +-inverses 48.5%

          \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

      16. metadata-eval 48.5%

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

      17. *-rgt-identity 48.5%

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

      18. associate-/l* 48.5%

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

      19. metadata-eval 48.5%

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

      20. *-commutative 48.5%

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

      21. neg-mul-1 48.5%

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

   8. Simplified 48.5%

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

   9. Taylor expanded in x around 0 67.2%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{-1 \cdot x}, x\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 67.2%

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

   11. Simplified 67.2%

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

   if 0.5 < x

   1. Initial program 58.2%

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

      1. +-commutative 58.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 44.6%

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

      1. log1p-define 44.6%

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

      2. rem-square-sqrt 44.6%

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

      3. fabs-sqr 44.6%

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

      4. rem-square-sqrt 44.6%

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

   7. Simplified 44.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

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

Alternative 12: 54.3% 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 37.2%

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

   1. +-commutative 37.2%

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

   2. hypot-1-def 59.5%

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

3. Simplified 59.5%

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

   1. flip-+ 12.8%

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

   2. clear-num 12.9%

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

   3. log-div 13.1%

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

   4. metadata-eval 13.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   5. add-sqr-sqrt 5.6%

      \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   6. fabs-sqr 5.6%

      \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   7. add-sqr-sqrt 13.1%

      \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   8. pow2 13.1%

      \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   9. add-sqr-sqrt 5.8%

      \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   10. fabs-sqr 5.8%

       \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   11. add-sqr-sqrt 13.1%

       \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   12. hypot-1-def 13.2%

       \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   13. hypot-1-def 13.1%

       \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

   14. add-sqr-sqrt 13.4%

       \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   15. +-commutative 13.4%

       \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

6. Applied egg-rr 13.4%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
7. Step-by-step derivation

   1. neg-sub0 13.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

   2. div-sub 13.4%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{x}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]

   3. fma-undefine 13.4%

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

   4. unpow2 13.4%

      \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

   5. associate--r+ 13.4%

      \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

   6. +-inverses 13.4%

      \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

   7. metadata-eval 13.4%

      \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

   8. *-rgt-identity 13.4%

      \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x \cdot 1}}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

   9. associate-/l* 13.4%

      \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot \frac{1}{-1}} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

   10. metadata-eval 13.4%

       \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

   11. *-commutative 13.4%

       \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{-1 \cdot x} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

   12. fma-undefine 13.4%

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

   13. unpow2 13.4%

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

   14. associate--r+ 25.1%

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

   15. +-inverses 39.8%

       \[\leadsto \mathsf{copysign}\left(-\log \left(-1 \cdot x - \frac{\mathsf{hypot}\left(1, x\right)}{\color{blue}{0} - 1}\right), x\right) \]

   16. metadata-eval 39.8%

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

   17. *-rgt-identity 39.8%

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

   18. associate-/l* 39.8%

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

   19. metadata-eval 39.8%

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

   20. *-commutative 39.8%

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

   21. neg-mul-1 39.8%

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

8. Simplified 39.8%

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

9. Taylor expanded in x around 0 54.8%

   \[\leadsto \mathsf{copysign}\left(-\color{blue}{-1 \cdot x}, x\right) \]
10. Step-by-step derivation

    1. neg-mul-1 54.8%

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

11. Simplified 54.8%

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

12. Final simplification 54.8%

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

Developer target: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024073 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))