Rust f32::asinh

Percentage Accurate: 37.0% → 99.5%

Time: 8.1s

Alternatives: 13

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.11999999731779099:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right) \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.11999999731779099)
     (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.10000000149011612)
       (copysign
        (+ x (* (fma (pow x 2.0) 0.075 -0.16666666666666666) (pow x 3.0)))
        x)
       (copysign (log (+ (fabs 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 <= -0.11999999731779099f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = copysignf((x + (fmaf(powf(x, 2.0f), 0.075f, -0.16666666666666666f) * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((fabsf(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(-0.11999999731779099))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x + Float32(fma((x ^ Float32(2.0)), Float32(0.075), Float32(-0.16666666666666666)) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(abs(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) < -0.119999997

   1. Initial program 47.1%

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

      1. +-commutative 47.1%

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

      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. flip-+ 4.9%

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

      4. hypot-1-def 4.9%

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

      5. hypot-1-def 4.9%

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

      6. add-sqr-sqrt 4.6%

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

      7. +-commutative 4.6%

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

      8. hypot-1-def 4.6%

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

      9. +-commutative 4.6%

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

      10. div-sub 4.8%

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

   4. Applied egg-rr 7.0%

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

      1. div-sub 8.8%

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

      2. remove-double-neg 8.8%

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

      3. distribute-neg-frac2 8.8%

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

      4. distribute-frac-neg 8.8%

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

      5. neg-sub0 8.8%

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

      6. associate--r- 8.8%

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

      7. neg-sub0 8.8%

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

      8. +-commutative 8.8%

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

      9. fma-undefine 8.8%

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

      10. unpow2 8.8%

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

      11. +-commutative 8.8%

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

      12. associate-+l+ 43.5%

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

      13. sub-neg 43.5%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -0.119999997 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.100000001

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

      2. hypot-1-def 21.4%

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

      3. flip-+ 21.7%

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

      4. hypot-1-def 21.7%

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

      5. hypot-1-def 21.6%

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

      6. add-sqr-sqrt 21.7%

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

      7. +-commutative 21.7%

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

      8. hypot-1-def 21.6%

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

      9. +-commutative 21.6%

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

      10. div-sub 21.7%

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

   4. Applied egg-rr 21.8%

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

      1. div-sub 21.7%

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

      2. remove-double-neg 21.7%

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

      3. distribute-neg-frac2 21.7%

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

      4. distribute-frac-neg 21.7%

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

      5. neg-sub0 21.7%

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

      6. associate--r- 21.7%

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

      7. neg-sub0 21.7%

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

      8. +-commutative 21.7%

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

      9. fma-undefine 21.7%

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

      10. unpow2 21.7%

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

      11. +-commutative 21.7%

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

      12. associate-+l+ 21.8%

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

      13. sub-neg 21.8%

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

      14. +-inverses 21.8%

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

      15. metadata-eval 21.8%

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

      16. neg-sub0 21.8%

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

      17. associate--r- 21.8%

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

      18. neg-sub0 21.8%

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

      19. +-commutative 21.8%

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

      20. sub-neg 21.8%

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

   6. Simplified 21.8%

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

   7. Taylor expanded in x around 0 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. fmm-def 100.0%

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

      7. metadata-eval 100.0%

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

      8. unpow2 100.0%

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

      9. unpow3 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}}, x\right) \]

   9. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + \mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right) \cdot {x}^{3}}, x\right) \]

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

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. hypot-1-def 98.6%

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

   3. Simplified 98.6%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.11999999731779099:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.11999999731779099)
     (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.10000000149011612)
       (copysign
        (*
         x
         (+ 1.0 (* (pow x 2.0) (- (* (pow x 2.0) 0.075) 0.16666666666666666))))
        x)
       (copysign (log (+ (fabs 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 <= -0.11999999731779099f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((powf(x, 2.0f) * 0.075f) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((fabsf(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(-0.11999999731779099))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(0.075)) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(abs(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(-0.11999999731779099))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (t_0 <= single(0.10000000149011612))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x ^ single(2.0)) * single(0.075)) - single(0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((abs(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) < -0.119999997

   1. Initial program 47.1%

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

      1. +-commutative 47.1%

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

      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. flip-+ 4.9%

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

      4. hypot-1-def 4.9%

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

      5. hypot-1-def 4.9%

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

      6. add-sqr-sqrt 4.6%

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

      7. +-commutative 4.6%

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

      8. hypot-1-def 4.6%

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

      9. +-commutative 4.6%

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

      10. div-sub 4.8%

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

   4. Applied egg-rr 7.0%

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

      1. div-sub 8.8%

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

      2. remove-double-neg 8.8%

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

      3. distribute-neg-frac2 8.8%

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

      4. distribute-frac-neg 8.8%

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

      5. neg-sub0 8.8%

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

      6. associate--r- 8.8%

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

      7. neg-sub0 8.8%

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

      8. +-commutative 8.8%

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

      9. fma-undefine 8.8%

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

      10. unpow2 8.8%

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

      11. +-commutative 8.8%

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

      12. associate-+l+ 43.5%

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

      13. sub-neg 43.5%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -0.119999997 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.100000001

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

      2. hypot-1-def 21.4%

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

      3. flip-+ 21.7%

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

      4. hypot-1-def 21.7%

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

      5. hypot-1-def 21.6%

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

      6. add-sqr-sqrt 21.7%

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

      7. +-commutative 21.7%

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

      8. hypot-1-def 21.6%

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

      9. +-commutative 21.6%

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

      10. div-sub 21.7%

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

   4. Applied egg-rr 21.8%

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

      1. div-sub 21.7%

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

      2. remove-double-neg 21.7%

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

      3. distribute-neg-frac2 21.7%

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

      4. distribute-frac-neg 21.7%

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

      5. neg-sub0 21.7%

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

      6. associate--r- 21.7%

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

      7. neg-sub0 21.7%

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

      8. +-commutative 21.7%

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

      9. fma-undefine 21.7%

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

      10. unpow2 21.7%

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

      11. +-commutative 21.7%

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

      12. associate-+l+ 21.8%

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

      13. sub-neg 21.8%

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

      14. +-inverses 21.8%

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

      15. metadata-eval 21.8%

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

      16. neg-sub0 21.8%

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

      17. associate--r- 21.8%

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

      18. neg-sub0 21.8%

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

      19. +-commutative 21.8%

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

      20. sub-neg 21.8%

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

   6. Simplified 21.8%

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

   7. Taylor expanded in x around 0 99.9%

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

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

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. hypot-1-def 98.6%

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

   3. Simplified 98.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.11999999731779099:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.11999999731779099:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.11999999731779099)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 0.10000000149011612)
     (copysign
      (*
       x
       (+ 1.0 (* (pow x 2.0) (- (* (pow x 2.0) 0.075) 0.16666666666666666))))
      x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.11999999731779099f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (x <= 0.10000000149011612f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((powf(x, 2.0f) * 0.075f) - 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(-0.11999999731779099))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(0.075)) - 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(-0.11999999731779099))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (x <= single(0.10000000149011612))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x ^ single(2.0)) * single(0.075)) - 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 < -0.119999997

   1. Initial program 47.1%

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

      1. +-commutative 47.1%

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

      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. flip-+ 4.9%

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

      4. hypot-1-def 4.9%

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

      5. hypot-1-def 4.9%

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

      6. add-sqr-sqrt 4.6%

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

      7. +-commutative 4.6%

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

      8. hypot-1-def 4.6%

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

      9. +-commutative 4.6%

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

      10. div-sub 4.8%

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

   4. Applied egg-rr 7.0%

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

      1. div-sub 8.8%

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

      2. remove-double-neg 8.8%

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

      3. distribute-neg-frac2 8.8%

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

      4. distribute-frac-neg 8.8%

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

      5. neg-sub0 8.8%

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

      6. associate--r- 8.8%

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

      7. neg-sub0 8.8%

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

      8. +-commutative 8.8%

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

      9. fma-undefine 8.8%

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

      10. unpow2 8.8%

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

      11. +-commutative 8.8%

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

      12. associate-+l+ 43.5%

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

      13. sub-neg 43.5%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -0.119999997 < x < 0.100000001

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

      2. hypot-1-def 21.4%

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

      3. flip-+ 21.7%

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

      4. hypot-1-def 21.7%

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

      5. hypot-1-def 21.6%

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

      6. add-sqr-sqrt 21.7%

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

      7. +-commutative 21.7%

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

      8. hypot-1-def 21.6%

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

      9. +-commutative 21.6%

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

      10. div-sub 21.7%

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

   4. Applied egg-rr 21.8%

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

      1. div-sub 21.7%

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

      2. remove-double-neg 21.7%

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

      3. distribute-neg-frac2 21.7%

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

      4. distribute-frac-neg 21.7%

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

      5. neg-sub0 21.7%

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

      6. associate--r- 21.7%

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

      7. neg-sub0 21.7%

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

      8. +-commutative 21.7%

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

      9. fma-undefine 21.7%

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

      10. unpow2 21.7%

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

      11. +-commutative 21.7%

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

      12. associate-+l+ 21.8%

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

      13. sub-neg 21.8%

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

      14. +-inverses 21.8%

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

      15. metadata-eval 21.8%

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

      16. neg-sub0 21.8%

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

      17. associate--r- 21.8%

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

      18. neg-sub0 21.8%

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

      19. +-commutative 21.8%

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

      20. sub-neg 21.8%

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

   6. Simplified 21.8%

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

   7. Taylor expanded in x around 0 99.9%

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

   if 0.100000001 < x

   1. Initial program 41.7%

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

   3. Taylor expanded in x around 0 41.7%

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

      1. rem-square-sqrt 41.7%

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

      2. fabs-sqr 41.7%

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

      3. metadata-eval 41.7%

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

      4. unpow2 41.7%

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

      5. hypot-undefine 98.6%

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

      6. rem-square-sqrt 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.11999999731779099:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.10000000149011612)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.10000000149011612f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.10000000149011612))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), 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(-0.10000000149011612))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	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 < -0.100000001

   1. Initial program 47.8%

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

      1. +-commutative 47.8%

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

      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. flip-+ 6.3%

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

      4. hypot-1-def 6.3%

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

      5. hypot-1-def 6.3%

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

      6. add-sqr-sqrt 6.0%

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

      7. +-commutative 6.0%

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

      8. hypot-1-def 6.0%

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

      9. +-commutative 6.0%

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

      10. div-sub 6.2%

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

   4. Applied egg-rr 8.4%

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

      1. div-sub 10.2%

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

      2. remove-double-neg 10.2%

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

      3. distribute-neg-frac2 10.2%

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

      4. distribute-frac-neg 10.2%

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

      5. neg-sub0 10.2%

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

      6. associate--r- 10.2%

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

      7. neg-sub0 10.2%

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

      8. +-commutative 10.2%

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

      9. fma-undefine 10.2%

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

      10. unpow2 10.2%

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

      11. +-commutative 10.2%

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

      12. associate-+l+ 44.3%

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

      13. sub-neg 44.3%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -0.100000001 < x < 0.0500000007

   1. Initial program 20.2%

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

      1. +-commutative 20.2%

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

      2. hypot-1-def 20.2%

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

      3. flip-+ 20.5%

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

      4. hypot-1-def 20.5%

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

      5. hypot-1-def 20.5%

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

      6. add-sqr-sqrt 20.5%

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

      7. +-commutative 20.5%

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

      8. hypot-1-def 20.5%

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

      9. +-commutative 20.5%

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

      10. div-sub 20.6%

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

   4. Applied egg-rr 20.6%

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

      1. div-sub 20.5%

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

      2. remove-double-neg 20.5%

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

      3. distribute-neg-frac2 20.5%

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

      4. distribute-frac-neg 20.5%

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

      5. neg-sub0 20.5%

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

      6. associate--r- 20.5%

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

      7. neg-sub0 20.5%

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

      8. +-commutative 20.5%

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

      9. fma-undefine 20.5%

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

      10. unpow2 20.5%

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

      11. +-commutative 20.5%

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

      12. associate-+l+ 20.6%

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

      13. sub-neg 20.6%

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

      14. +-inverses 20.6%

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

      15. metadata-eval 20.6%

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

      16. neg-sub0 20.6%

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

      17. associate--r- 20.6%

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

      18. neg-sub0 20.6%

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

      19. +-commutative 20.6%

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

      20. sub-neg 20.6%

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

   6. Simplified 20.6%

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

   7. Taylor expanded in x around 0 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. *-lft-identity 99.9%

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

      3. associate-*l* 99.9%

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

      4. unpow2 99.9%

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

      5. unpow3 99.9%

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

   9. Simplified 99.9%

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

   if 0.0500000007 < x

   1. Initial program 42.5%

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

   3. Taylor expanded in x around 0 42.5%

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

      1. rem-square-sqrt 42.5%

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

      2. fabs-sqr 42.5%

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

      3. metadata-eval 42.5%

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

      4. unpow2 42.5%

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

      5. hypot-undefine 98.5%

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

      6. rem-square-sqrt 98.6%

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

   5. Simplified 98.6%

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

Alternative 5: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.10000000149011612)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.10000000149011612f) {
		tmp = copysignf(logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.10000000149011612))
		tmp = copysign(log(Float32(hypot(Float32(1.0), x) - x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), 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(-0.10000000149011612))
		tmp = sign(x) * abs(log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	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 < -0.100000001

   1. Initial program 47.8%

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

      1. +-commutative 47.8%

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

      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. flip-+ 6.3%

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

      4. hypot-1-def 6.3%

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

      5. hypot-1-def 6.3%

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

      6. add-sqr-sqrt 6.0%

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

      7. +-commutative 6.0%

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

      8. hypot-1-def 6.0%

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

      9. +-commutative 6.0%

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

      10. div-sub 6.2%

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

   4. Applied egg-rr 8.4%

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

      1. div-sub 10.2%

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

      2. remove-double-neg 10.2%

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

      3. distribute-neg-frac2 10.2%

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

      4. distribute-frac-neg 10.2%

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

      5. neg-sub0 10.2%

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

      6. associate--r- 10.2%

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

      7. neg-sub0 10.2%

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

      8. +-commutative 10.2%

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

      9. fma-undefine 10.2%

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

      10. unpow2 10.2%

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

      11. +-commutative 10.2%

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

      12. associate-+l+ 44.3%

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

      13. sub-neg 44.3%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

      1. expm1-log1p-u 3.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 3.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\right)} - 1}, x\right) \]

      3. log-rec 3.0%

         \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}\right)} - 1, x\right) \]

   8. Applied egg-rr 3.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} - 1}, x\right) \]
   9. Step-by-step derivation

      1. sub-neg 3.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} + \left(-1\right)}, x\right) \]

      2. metadata-eval 3.0%

         \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} + \color{blue}{-1}, x\right) \]

      3. +-commutative 3.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 + e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}}, x\right) \]

      4. log1p-undefine 3.0%

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

      5. rem-exp-log 99.8%

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

      6. unsub-neg 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 47.8%

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

       1. metadata-eval 47.8%

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

       2. unpow2 47.8%

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

       3. hypot-undefine 99.8%

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

       4. rem-square-sqrt -0.0%

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

       5. fabs-sqr -0.0%

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

       6. rem-square-sqrt 99.8%

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

       7. fabs-neg 99.8%

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

       8. rem-square-sqrt 98.4%

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

       9. fabs-sqr 98.4%

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

       10. rem-square-sqrt 99.8%

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

   13. Simplified 99.8%

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

   if -0.100000001 < x < 0.0500000007

   1. Initial program 20.2%

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

      1. +-commutative 20.2%

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

      2. hypot-1-def 20.2%

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

      3. flip-+ 20.5%

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

      4. hypot-1-def 20.5%

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

      5. hypot-1-def 20.5%

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

      6. add-sqr-sqrt 20.5%

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

      7. +-commutative 20.5%

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

      8. hypot-1-def 20.5%

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

      9. +-commutative 20.5%

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

      10. div-sub 20.6%

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

   4. Applied egg-rr 20.6%

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

      1. div-sub 20.5%

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

      2. remove-double-neg 20.5%

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

      3. distribute-neg-frac2 20.5%

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

      4. distribute-frac-neg 20.5%

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

      5. neg-sub0 20.5%

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

      6. associate--r- 20.5%

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

      7. neg-sub0 20.5%

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

      8. +-commutative 20.5%

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

      9. fma-undefine 20.5%

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

      10. unpow2 20.5%

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

      11. +-commutative 20.5%

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

      12. associate-+l+ 20.6%

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

      13. sub-neg 20.6%

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

      14. +-inverses 20.6%

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

      15. metadata-eval 20.6%

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

      16. neg-sub0 20.6%

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

      17. associate--r- 20.6%

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

      18. neg-sub0 20.6%

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

      19. +-commutative 20.6%

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

      20. sub-neg 20.6%

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

   6. Simplified 20.6%

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

   7. Taylor expanded in x around 0 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. *-lft-identity 99.9%

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

      3. associate-*l* 99.9%

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

      4. unpow2 99.9%

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

      5. unpow3 99.9%

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

   9. Simplified 99.9%

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

   if 0.0500000007 < x

   1. Initial program 42.5%

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

   3. Taylor expanded in x around 0 42.5%

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

      1. rem-square-sqrt 42.5%

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

      2. fabs-sqr 42.5%

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

      3. metadata-eval 42.5%

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

      4. unpow2 42.5%

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

      5. hypot-undefine 98.5%

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

      6. rem-square-sqrt 98.6%

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

   5. Simplified 98.6%

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

Alternative 6: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), 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(-50.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	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 < -50

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      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. flip-+ 3.4%

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

      4. hypot-1-def 3.4%

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

      5. hypot-1-def 3.4%

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

      6. add-sqr-sqrt 3.2%

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

      7. +-commutative 3.2%

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

      8. hypot-1-def 3.2%

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

      9. +-commutative 3.2%

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

      10. div-sub 3.3%

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

   4. Applied egg-rr 5.5%

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

      1. div-sub 7.3%

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

      2. remove-double-neg 7.3%

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

      3. distribute-neg-frac2 7.3%

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

      4. distribute-frac-neg 7.3%

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

      5. neg-sub0 7.3%

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

      6. associate--r- 7.3%

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

      7. neg-sub0 7.3%

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

      8. +-commutative 7.3%

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

      9. fma-undefine 7.3%

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

      10. unpow2 7.3%

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

      11. +-commutative 7.3%

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

      12. associate-+l+ 42.6%

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

      13. sub-neg 42.6%

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

      14. +-inverses 100.0%

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

      15. metadata-eval 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. neg-sub0 100.0%

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

      19. +-commutative 100.0%

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

      20. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around -inf 99.4%

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

   if -50 < x < 0.0500000007

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

      2. hypot-1-def 21.4%

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

      3. flip-+ 21.7%

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

      4. hypot-1-def 21.7%

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

      5. hypot-1-def 21.7%

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

      6. add-sqr-sqrt 21.6%

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

      7. +-commutative 21.6%

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

      8. hypot-1-def 21.6%

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

      9. +-commutative 21.6%

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

      10. div-sub 21.8%

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

   4. Applied egg-rr 21.8%

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

      1. div-sub 21.7%

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

      2. remove-double-neg 21.7%

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

      3. distribute-neg-frac2 21.7%

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

      4. distribute-frac-neg 21.7%

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

      5. neg-sub0 21.7%

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

      6. associate--r- 21.7%

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

      7. neg-sub0 21.7%

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

      8. +-commutative 21.7%

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

      9. fma-undefine 21.7%

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

      10. unpow2 21.7%

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

      11. +-commutative 21.7%

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

      12. associate-+l+ 21.8%

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

      13. sub-neg 21.8%

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

      14. +-inverses 21.8%

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

      15. metadata-eval 21.8%

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

      16. neg-sub0 21.8%

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

      17. associate--r- 21.8%

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

      18. neg-sub0 21.8%

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

      19. +-commutative 21.8%

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

      20. sub-neg 21.8%

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

   6. Simplified 21.8%

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

   7. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   8. 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. associate-*l* 99.5%

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

      4. unpow2 99.5%

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

      5. unpow3 99.5%

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

   9. Simplified 99.5%

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

   if 0.0500000007 < x

   1. Initial program 42.5%

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

   3. Taylor expanded in x around 0 42.5%

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

      1. rem-square-sqrt 42.5%

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

      2. fabs-sqr 42.5%

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

      3. metadata-eval 42.5%

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

      4. unpow2 42.5%

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

      5. hypot-undefine 98.5%

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

      6. rem-square-sqrt 98.6%

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

   5. Simplified 98.6%

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

Alternative 7: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.5)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.5f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.5))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-50.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.5))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	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 < -50

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      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. flip-+ 3.4%

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

      4. hypot-1-def 3.4%

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

      5. hypot-1-def 3.4%

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

      6. add-sqr-sqrt 3.2%

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

      7. +-commutative 3.2%

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

      8. hypot-1-def 3.2%

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

      9. +-commutative 3.2%

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

      10. div-sub 3.3%

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

   4. Applied egg-rr 5.5%

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

      1. div-sub 7.3%

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

      2. remove-double-neg 7.3%

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

      3. distribute-neg-frac2 7.3%

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

      4. distribute-frac-neg 7.3%

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

      5. neg-sub0 7.3%

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

      6. associate--r- 7.3%

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

      7. neg-sub0 7.3%

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

      8. +-commutative 7.3%

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

      9. fma-undefine 7.3%

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

      10. unpow2 7.3%

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

      11. +-commutative 7.3%

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

      12. associate-+l+ 42.6%

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

      13. sub-neg 42.6%

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

      14. +-inverses 100.0%

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

      15. metadata-eval 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. neg-sub0 100.0%

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

      19. +-commutative 100.0%

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

      20. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around -inf 99.4%

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

   if -50 < x < 1.5

   1. Initial program 22.6%

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

      1. +-commutative 22.6%

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

      2. hypot-1-def 22.6%

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

      3. flip-+ 22.8%

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

      4. hypot-1-def 22.8%

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

      5. hypot-1-def 22.8%

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

      6. add-sqr-sqrt 22.8%

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

      7. +-commutative 22.8%

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

      8. hypot-1-def 22.8%

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

      9. +-commutative 22.8%

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

      10. div-sub 22.9%

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

   4. Applied egg-rr 23.0%

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

      1. div-sub 22.9%

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

      2. remove-double-neg 22.9%

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

      3. distribute-neg-frac2 22.9%

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

      4. distribute-frac-neg 22.9%

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

      5. neg-sub0 22.9%

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

      6. associate--r- 22.9%

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

      7. neg-sub0 22.9%

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

      8. +-commutative 22.9%

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

      9. fma-undefine 22.9%

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

      10. unpow2 22.9%

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

      11. +-commutative 22.9%

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

      12. associate-+l+ 23.0%

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

      13. sub-neg 23.0%

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

      14. +-inverses 23.0%

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

      15. metadata-eval 23.0%

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

      16. neg-sub0 23.0%

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

      17. associate--r- 23.0%

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

      18. neg-sub0 23.0%

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

      19. +-commutative 23.0%

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

      20. sub-neg 23.0%

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

   6. Simplified 23.0%

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

   7. Taylor expanded in x around 0 98.9%

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

      1. distribute-rgt-in 98.9%

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

      2. *-lft-identity 98.9%

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

      3. associate-*l* 98.9%

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

      4. unpow2 98.9%

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

      5. unpow3 98.9%

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

   9. Simplified 98.9%

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

   if 1.5 < x

   1. Initial program 40.9%

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

      1. +-commutative 40.9%

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

      2. hypot-1-def 98.6%

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

      3. flip-+ 6.8%

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

      4. hypot-1-def 6.8%

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

      5. hypot-1-def 6.8%

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

      6. add-sqr-sqrt 6.7%

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

      7. +-commutative 6.7%

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

      8. hypot-1-def 6.7%

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

      9. +-commutative 6.7%

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

      10. div-sub 7.0%

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

   4. Applied egg-rr 6.9%

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

      1. div-sub 6.6%

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

      2. remove-double-neg 6.6%

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

      3. distribute-neg-frac2 6.6%

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

      4. distribute-frac-neg 6.6%

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

      5. neg-sub0 6.6%

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

      6. associate--r- 6.6%

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

      7. neg-sub0 6.6%

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

      8. +-commutative 6.6%

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

      9. fma-undefine 6.7%

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

      10. unpow2 6.7%

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

      11. +-commutative 6.7%

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

      12. associate-+l+ 6.6%

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

      13. sub-neg 6.6%

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

      14. +-inverses 6.6%

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

      15. metadata-eval 6.6%

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

      16. neg-sub0 12.5%

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

      17. associate--r- 12.5%

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

      18. neg-sub0 12.5%

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

      19. +-commutative 12.5%

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

      20. sub-neg 12.5%

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

   6. Simplified 12.5%

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

      1. expm1-log1p-u 12.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 12.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\right)} - 1}, x\right) \]

      3. log-rec 12.5%

         \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}\right)} - 1, x\right) \]

   8. Applied egg-rr 12.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} - 1}, x\right) \]
   9. Step-by-step derivation

      1. sub-neg 12.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} + \left(-1\right)}, x\right) \]

      2. metadata-eval 12.5%

         \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} + \color{blue}{-1}, x\right) \]

      3. +-commutative 12.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 + e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}}, x\right) \]

      4. log1p-undefine 12.5%

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

      5. rem-exp-log 12.5%

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

      6. unsub-neg 12.5%

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

   10. Simplified 12.5%

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

   11. Taylor expanded in x around 0 12.5%

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

       1. metadata-eval 12.5%

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

       2. unpow2 12.5%

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

       3. hypot-undefine 12.5%

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

       4. rem-square-sqrt 12.5%

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

       5. fabs-sqr 12.5%

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

       6. rem-square-sqrt 12.5%

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

       7. fabs-neg 12.5%

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

       8. rem-square-sqrt -0.0%

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

       9. fabs-sqr -0.0%

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

       10. rem-square-sqrt 12.5%

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

   13. Simplified 12.5%

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

   14. Taylor expanded in x around inf 97.3%

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

Alternative 8: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.5) (copysign x x) (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.5f) {
		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(-50.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-50.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.5))
		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 < -50

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      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. flip-+ 3.4%

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

      4. hypot-1-def 3.4%

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

      5. hypot-1-def 3.4%

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

      6. add-sqr-sqrt 3.2%

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

      7. +-commutative 3.2%

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

      8. hypot-1-def 3.2%

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

      9. +-commutative 3.2%

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

      10. div-sub 3.3%

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

   4. Applied egg-rr 5.5%

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

      1. div-sub 7.3%

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

      2. remove-double-neg 7.3%

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

      3. distribute-neg-frac2 7.3%

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

      4. distribute-frac-neg 7.3%

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

      5. neg-sub0 7.3%

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

      6. associate--r- 7.3%

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

      7. neg-sub0 7.3%

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

      8. +-commutative 7.3%

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

      9. fma-undefine 7.3%

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

      10. unpow2 7.3%

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

      11. +-commutative 7.3%

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

      12. associate-+l+ 42.6%

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

      13. sub-neg 42.6%

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

      14. +-inverses 100.0%

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

      15. metadata-eval 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. neg-sub0 100.0%

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

      19. +-commutative 100.0%

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

      20. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around -inf 99.4%

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

   if -50 < x < 1.5

   1. Initial program 22.6%

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

      1. +-commutative 22.6%

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

      2. hypot-1-def 22.6%

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

      3. flip-+ 22.8%

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

      4. hypot-1-def 22.8%

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

      5. hypot-1-def 22.8%

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

      6. add-sqr-sqrt 22.8%

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

      7. +-commutative 22.8%

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

      8. hypot-1-def 22.8%

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

      9. +-commutative 22.8%

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

      10. div-sub 22.9%

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

   4. Applied egg-rr 23.0%

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

      1. div-sub 22.9%

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

      2. remove-double-neg 22.9%

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

      3. distribute-neg-frac2 22.9%

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

      4. distribute-frac-neg 22.9%

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

      5. neg-sub0 22.9%

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

      6. associate--r- 22.9%

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

      7. neg-sub0 22.9%

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

      8. +-commutative 22.9%

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

      9. fma-undefine 22.9%

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

      10. unpow2 22.9%

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

      11. +-commutative 22.9%

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

      12. associate-+l+ 23.0%

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

      13. sub-neg 23.0%

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

      14. +-inverses 23.0%

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

      15. metadata-eval 23.0%

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

      16. neg-sub0 23.0%

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

      17. associate--r- 23.0%

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

      18. neg-sub0 23.0%

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

      19. +-commutative 23.0%

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

      20. sub-neg 23.0%

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

   6. Simplified 23.0%

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

   7. Taylor expanded in x around 0 96.9%

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

   if 1.5 < x

   1. Initial program 40.9%

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

      1. +-commutative 40.9%

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

      2. hypot-1-def 98.6%

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

      3. flip-+ 6.8%

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

      4. hypot-1-def 6.8%

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

      5. hypot-1-def 6.8%

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

      6. add-sqr-sqrt 6.7%

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

      7. +-commutative 6.7%

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

      8. hypot-1-def 6.7%

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

      9. +-commutative 6.7%

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

      10. div-sub 7.0%

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

   4. Applied egg-rr 6.9%

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

      1. div-sub 6.6%

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

      2. remove-double-neg 6.6%

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

      3. distribute-neg-frac2 6.6%

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

      4. distribute-frac-neg 6.6%

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

      5. neg-sub0 6.6%

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

      6. associate--r- 6.6%

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

      7. neg-sub0 6.6%

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

      8. +-commutative 6.6%

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

      9. fma-undefine 6.7%

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

      10. unpow2 6.7%

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

      11. +-commutative 6.7%

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

      12. associate-+l+ 6.6%

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

      13. sub-neg 6.6%

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

      14. +-inverses 6.6%

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

      15. metadata-eval 6.6%

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

      16. neg-sub0 12.5%

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

      17. associate--r- 12.5%

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

      18. neg-sub0 12.5%

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

      19. +-commutative 12.5%

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

      20. sub-neg 12.5%

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

   6. Simplified 12.5%

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

      1. expm1-log1p-u 12.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 12.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\right)} - 1}, x\right) \]

      3. log-rec 12.5%

         \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}\right)} - 1, x\right) \]

   8. Applied egg-rr 12.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} - 1}, x\right) \]
   9. Step-by-step derivation

      1. sub-neg 12.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} + \left(-1\right)}, x\right) \]

      2. metadata-eval 12.5%

         \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} + \color{blue}{-1}, x\right) \]

      3. +-commutative 12.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 + e^{\mathsf{log1p}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}}, x\right) \]

      4. log1p-undefine 12.5%

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

      5. rem-exp-log 12.5%

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

      6. unsub-neg 12.5%

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

   10. Simplified 12.5%

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

   11. Taylor expanded in x around 0 12.5%

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

       1. metadata-eval 12.5%

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

       2. unpow2 12.5%

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

       3. hypot-undefine 12.5%

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

       4. rem-square-sqrt 12.5%

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

       5. fabs-sqr 12.5%

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

       6. rem-square-sqrt 12.5%

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

       7. fabs-neg 12.5%

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

       8. rem-square-sqrt -0.0%

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

       9. fabs-sqr -0.0%

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

       10. rem-square-sqrt 12.5%

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

   13. Simplified 12.5%

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

   14. Taylor expanded in x around inf 97.3%

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

Alternative 9: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.5) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-50.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.5))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -50

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      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. flip-+ 3.4%

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

      4. hypot-1-def 3.4%

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

      5. hypot-1-def 3.4%

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

      6. add-sqr-sqrt 3.2%

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

      7. +-commutative 3.2%

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

      8. hypot-1-def 3.2%

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

      9. +-commutative 3.2%

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

      10. div-sub 3.3%

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

   4. Applied egg-rr 5.5%

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

      1. div-sub 7.3%

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

      2. remove-double-neg 7.3%

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

      3. distribute-neg-frac2 7.3%

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

      4. distribute-frac-neg 7.3%

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

      5. neg-sub0 7.3%

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

      6. associate--r- 7.3%

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

      7. neg-sub0 7.3%

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

      8. +-commutative 7.3%

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

      9. fma-undefine 7.3%

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

      10. unpow2 7.3%

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

      11. +-commutative 7.3%

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

      12. associate-+l+ 42.6%

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

      13. sub-neg 42.6%

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

      14. +-inverses 100.0%

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

      15. metadata-eval 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate--r- 100.0%

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

      18. neg-sub0 100.0%

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

      19. +-commutative 100.0%

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

      20. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around -inf 99.4%

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

   if -50 < x < 1.5

   1. Initial program 22.6%

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

      1. +-commutative 22.6%

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

      2. hypot-1-def 22.6%

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

      3. flip-+ 22.8%

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

      4. hypot-1-def 22.8%

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

      5. hypot-1-def 22.8%

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

      6. add-sqr-sqrt 22.8%

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

      7. +-commutative 22.8%

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

      8. hypot-1-def 22.8%

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

      9. +-commutative 22.8%

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

      10. div-sub 22.9%

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

   4. Applied egg-rr 23.0%

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

      1. div-sub 22.9%

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

      2. remove-double-neg 22.9%

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

      3. distribute-neg-frac2 22.9%

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

      4. distribute-frac-neg 22.9%

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

      5. neg-sub0 22.9%

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

      6. associate--r- 22.9%

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

      7. neg-sub0 22.9%

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

      8. +-commutative 22.9%

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

      9. fma-undefine 22.9%

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

      10. unpow2 22.9%

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

      11. +-commutative 22.9%

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

      12. associate-+l+ 23.0%

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

      13. sub-neg 23.0%

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

      14. +-inverses 23.0%

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

      15. metadata-eval 23.0%

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

      16. neg-sub0 23.0%

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

      17. associate--r- 23.0%

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

      18. neg-sub0 23.0%

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

      19. +-commutative 23.0%

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

      20. sub-neg 23.0%

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

   6. Simplified 23.0%

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

   7. Taylor expanded in x around 0 96.9%

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

   if 1.5 < x

   1. Initial program 40.9%

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

   3. Taylor expanded in x around inf 95.9%

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

      1. +-commutative 95.9%

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

      2. rem-square-sqrt 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{x} + 1\right)\right), x\right) \]

      3. fabs-sqr 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x} + 1\right)\right), x\right) \]

      4. rem-square-sqrt 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\color{blue}{x}}{x} + 1\right)\right), x\right) \]

      5. *-inverses 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\color{blue}{1} + 1\right)\right), x\right) \]

      6. metadata-eval 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \color{blue}{2}\right), x\right) \]

   5. Simplified 95.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (- x)) x)
   (if (<= x 1.5) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 1.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(1.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-50.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(1.5))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -50

   1. Initial program 46.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 44.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 44.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   5. Simplified 44.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   if -50 < x < 1.5

   1. Initial program 22.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 22.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]

      2. hypot-1-def 22.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      3. flip-+ 22.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

      4. hypot-1-def 22.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. hypot-1-def 22.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. add-sqr-sqrt 22.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\left(1 + x \cdot x\right)}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. +-commutative 22.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\left(x \cdot x + 1\right)}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. hypot-1-def 22.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \left(x \cdot x + 1\right)}{\left|x\right| - \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      9. +-commutative 22.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \left(x \cdot x + 1\right)}{\left|x\right| - \sqrt{\color{blue}{x \cdot x + 1}}}\right), x\right) \]

      10. div-sub 22.9%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right|}{\left|x\right| - \sqrt{x \cdot x + 1}} - \frac{x \cdot x + 1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

   4. Applied egg-rr 23.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. div-sub 22.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

      2. remove-double-neg 22.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{\color{blue}{-\left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}}\right), x\right) \]

      3. distribute-neg-frac2 22.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]

      4. distribute-frac-neg 22.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]

      5. neg-sub0 22.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      6. associate--r- 22.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      7. neg-sub0 22.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      8. +-commutative 22.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      9. fma-undefine 22.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(x \cdot x + 1\right)} + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      10. unpow2 22.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left(\color{blue}{{x}^{2}} + 1\right) + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      11. +-commutative 22.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(1 + {x}^{2}\right)} + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      12. associate-+l+ 23.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1 + \left({x}^{2} + \left(-{x}^{2}\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      13. sub-neg 23.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1 + \color{blue}{\left({x}^{2} - {x}^{2}\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      14. +-inverses 23.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1 + \color{blue}{0}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      15. metadata-eval 23.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      16. neg-sub0 23.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

      17. associate--r- 23.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

      18. neg-sub0 23.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      19. +-commutative 23.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right), x\right) \]

      20. sub-neg 23.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

   6. Simplified 23.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]

   7. Taylor expanded in x around 0 96.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 1.5 < x

   1. Initial program 40.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 95.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \color{blue}{\left(\frac{\left|x\right|}{x} + 1\right)}\right), x\right) \]

      2. rem-square-sqrt 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{x} + 1\right)\right), x\right) \]

      3. fabs-sqr 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x} + 1\right)\right), x\right) \]

      4. rem-square-sqrt 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\frac{\color{blue}{x}}{x} + 1\right)\right), x\right) \]

      5. *-inverses 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\color{blue}{1} + 1\right)\right), x\right) \]

      6. metadata-eval 95.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \color{blue}{2}\right), x\right) \]

   5. Simplified 95.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 68.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0) (copysign (log (- x)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50.0))
		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 < -50

   1. Initial program 46.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 44.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 44.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   5. Simplified 44.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   if -50 < x

   1. Initial program 29.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 28.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define 76.6%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

      2. rem-square-sqrt 44.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

      3. fabs-sqr 44.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]

      4. rem-square-sqrt 76.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right), x\right) \]

   5. Simplified 76.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 63.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 1.5) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 1.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 30.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 30.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]

      2. hypot-1-def 47.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      3. flip-+ 16.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

      4. hypot-1-def 16.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. hypot-1-def 16.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. add-sqr-sqrt 16.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\left(1 + x \cdot x\right)}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. +-commutative 16.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\left(x \cdot x + 1\right)}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. hypot-1-def 16.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \left(x \cdot x + 1\right)}{\left|x\right| - \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      9. +-commutative 16.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \left(x \cdot x + 1\right)}{\left|x\right| - \sqrt{\color{blue}{x \cdot x + 1}}}\right), x\right) \]

      10. div-sub 16.5%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right|}{\left|x\right| - \sqrt{x \cdot x + 1}} - \frac{x \cdot x + 1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

   4. Applied egg-rr 17.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. div-sub 17.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

      2. remove-double-neg 17.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{\color{blue}{-\left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}}\right), x\right) \]

      3. distribute-neg-frac2 17.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]

      4. distribute-frac-neg 17.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]

      5. neg-sub0 17.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      6. associate--r- 17.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      7. neg-sub0 17.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      8. +-commutative 17.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      9. fma-undefine 17.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(x \cdot x + 1\right)} + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      10. unpow2 17.8%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left(\color{blue}{{x}^{2}} + 1\right) + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      11. +-commutative 17.8%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(1 + {x}^{2}\right)} + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      12. associate-+l+ 29.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1 + \left({x}^{2} + \left(-{x}^{2}\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      13. sub-neg 29.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1 + \color{blue}{\left({x}^{2} - {x}^{2}\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      14. +-inverses 48.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1 + \color{blue}{0}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      15. metadata-eval 48.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      16. neg-sub0 48.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

      17. associate--r- 48.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

      18. neg-sub0 48.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      19. +-commutative 48.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right), x\right) \]

      20. sub-neg 48.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

   6. Simplified 48.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]

   7. Taylor expanded in x around 0 68.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 1.5 < x

   1. Initial program 40.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 45.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define 45.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

      2. rem-square-sqrt 45.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

      3. fabs-sqr 45.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]

      4. rem-square-sqrt 45.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right), x\right) \]

   5. Simplified 45.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 54.4% 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 33.1%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 33.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]

   2. hypot-1-def 61.2%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. flip-+ 13.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   4. hypot-1-def 13.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   5. hypot-1-def 13.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   6. add-sqr-sqrt 13.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\left(1 + x \cdot x\right)}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   7. +-commutative 13.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\left(x \cdot x + 1\right)}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   8. hypot-1-def 13.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \left(x \cdot x + 1\right)}{\left|x\right| - \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

   9. +-commutative 13.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \left(x \cdot x + 1\right)}{\left|x\right| - \sqrt{\color{blue}{x \cdot x + 1}}}\right), x\right) \]

   10. div-sub 14.0%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right|}{\left|x\right| - \sqrt{x \cdot x + 1}} - \frac{x \cdot x + 1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

4. Applied egg-rr 14.5%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
5. Step-by-step derivation

   1. div-sub 14.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   2. remove-double-neg 14.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{\color{blue}{-\left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}}\right), x\right) \]

   3. distribute-neg-frac2 14.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]

   4. distribute-frac-neg 14.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]

   5. neg-sub0 14.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. associate--r- 14.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   7. neg-sub0 14.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   8. +-commutative 14.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   9. fma-undefine 14.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(x \cdot x + 1\right)} + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   10. unpow2 14.9%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left(\color{blue}{{x}^{2}} + 1\right) + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   11. +-commutative 14.9%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(1 + {x}^{2}\right)} + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   12. associate-+l+ 23.3%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1 + \left({x}^{2} + \left(-{x}^{2}\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   13. sub-neg 23.3%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1 + \color{blue}{\left({x}^{2} - {x}^{2}\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   14. +-inverses 37.0%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1 + \color{blue}{0}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   15. metadata-eval 37.0%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   16. neg-sub0 38.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

   17. associate--r- 38.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

   18. neg-sub0 38.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   19. +-commutative 38.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right), x\right) \]

   20. sub-neg 38.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

6. Simplified 38.5%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]

7. Taylor expanded in x around 0 53.4%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
8. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

float code(float x) {
	float t_0 = 1.0f / fabsf(x);
	return copysignf(log1pf((fabsf(x) + (fabsf(x) / (hypotf(1.0f, t_0) + t_0)))), x);
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) / abs(x))
	return copysign(log1p(Float32(abs(x) + Float32(abs(x) / Float32(hypot(Float32(1.0), t_0) + t_0)))), x)
end

Reproduce

herbie shell --seed 2024165 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (! :herbie-platform default (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))