Rust f32::asinh

Percentage Accurate: 38.3% → 99.5%

Time: 9.2s

Alternatives: 11

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 38.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.05000000074505806)
     (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.05000000074505806f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + (x + (0.075f * powf(x, 5.0f)))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.05000000074505806))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(x + Float32(Float32(0.075) * (x ^ Float32(5.0))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.05000000074505806))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (t_0 <= single(0.05000000074505806))
		tmp = sign(x) * abs(((single(-0.16666666666666666) * (x ^ single(3.0))) + (x + (single(0.075) * (x ^ single(5.0))))));
	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 (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -0.0500000007

   1. Initial program 53.9%

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

      1. +-commutative 53.9%

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

      2. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 15.0%

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

      2. div-sub 14.8%

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

      3. pow2 14.8%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 14.8%

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

      7. pow2 14.8%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 5.3%

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

      11. hypot-udef 5.3%

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

      12. hypot-udef 5.3%

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

      13. add-sqr-sqrt 5.3%

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

      14. metadata-eval 5.3%

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

      15. add-sqr-sqrt -0.0%

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

      16. fabs-sqr -0.0%

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

      17. add-sqr-sqrt 17.2%

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

   5. Applied egg-rr 17.2%

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

      1. unpow2 17.2%

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

      2. div-sub 19.1%

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

      3. unpow2 19.1%

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

      4. unpow2 19.1%

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

      5. unpow2 19.1%

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

      6. +-commutative 19.1%

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

      7. associate--r+ 50.8%

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

      8. +-inverses 99.9%

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

      9. metadata-eval 99.9%

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

      10. metadata-eval 99.9%

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

      11. associate-/r* 99.9%

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

      12. neg-mul-1 99.9%

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

      13. sub-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. distribute-neg-in 99.9%

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

      16. remove-double-neg 99.9%

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

      17. sub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.0500000007

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

      2. hypot-1-def 19.6%

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

   3. Simplified 19.6%

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

      1. flip-+ 19.5%

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

      2. clear-num 19.5%

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

      3. add-sqr-sqrt 11.4%

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

      4. fabs-sqr 11.4%

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

      5. add-sqr-sqrt 19.7%

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

      6. pow2 19.7%

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

      7. add-sqr-sqrt 11.4%

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

      8. fabs-sqr 11.4%

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

      9. add-sqr-sqrt 19.7%

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

      10. pow2 19.7%

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

      11. hypot-udef 19.7%

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

      12. hypot-udef 19.6%

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

      13. add-sqr-sqrt 19.7%

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

      14. metadata-eval 19.7%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)}, x\right) \]

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 50.8%

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

      1. +-commutative 50.8%

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

      2. hypot-1-def 98.8%

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

   3. Simplified 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. log-prod 98.8%

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

      3. metadata-eval 98.8%

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

      4. *-un-lft-identity 98.8%

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

      5. *-un-lft-identity 98.8%

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

      6. add-sqr-sqrt 98.7%

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

      7. fabs-sqr 98.7%

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

      8. add-sqr-sqrt 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. +-lft-identity 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.5)
   (copysign (- (log (* x -2.0))) 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 <= -1.5f) {
		tmp = copysignf(-logf((x * -2.0f)), 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(-1.5))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), 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(-1.5))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	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 < -1.5

   1. Initial program 49.3%

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

      1. +-commutative 49.3%

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

      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. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 98.9%

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

      1. associate--l+ 98.8%

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

      2. unpow1 98.8%

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

      3. sqr-pow -0.0%

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

      4. fabs-sqr -0.0%

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

      5. sqr-pow 10.8%

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

      6. unpow1 10.8%

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

      7. associate-+r- 97.5%

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

      8. mul-1-neg 97.5%

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

      9. sub-neg 97.5%

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

      10. +-inverses 97.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      11. neg-sub0 97.5%

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

      12. associate-*r/ 97.5%

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

      13. metadata-eval 97.5%

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

      14. distribute-neg-frac 97.5%

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

      15. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

      1. clear-num 97.5%

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

      2. log-div 97.5%

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

      3. metadata-eval 97.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{x}{-0.5}\right), x\right) \]

      4. div-inv 97.5%

         \[\leadsto \mathsf{copysign}\left(0 - \log \color{blue}{\left(x \cdot \frac{1}{-0.5}\right)}, x\right) \]

      5. metadata-eval 97.5%

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

   8. Applied egg-rr 97.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(x \cdot -2\right)}, x\right) \]
   9. Step-by-step derivation

      1. neg-sub0 97.5%

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

   10. Simplified 97.5%

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

   if -1.5 < x < 0.0500000007

   1. Initial program 22.5%

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

      1. +-commutative 22.5%

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

      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. Simplified 22.6%

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

      1. flip-+ 22.4%

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

      2. clear-num 22.4%

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

      3. add-sqr-sqrt 11.0%

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

      4. fabs-sqr 11.0%

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

      5. add-sqr-sqrt 22.7%

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

      6. pow2 22.7%

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

      7. add-sqr-sqrt 11.0%

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

      8. fabs-sqr 11.0%

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

      9. add-sqr-sqrt 22.7%

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

      10. pow2 22.7%

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

      11. hypot-udef 22.7%

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

      12. hypot-udef 22.6%

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

      13. add-sqr-sqrt 22.8%

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

      14. metadata-eval 22.8%

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

   5. Applied egg-rr 22.8%

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

   6. Taylor expanded in x around 0 98.2%

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

   if 0.0500000007 < x

   1. Initial program 50.8%

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

      1. +-commutative 50.8%

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

      2. hypot-1-def 98.8%

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

   3. Simplified 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. log-prod 98.8%

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

      3. metadata-eval 98.8%

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

      4. *-un-lft-identity 98.8%

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

      5. *-un-lft-identity 98.8%

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

      6. add-sqr-sqrt 98.7%

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

      7. fabs-sqr 98.7%

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

      8. add-sqr-sqrt 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. +-lft-identity 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05000000074505806)
   (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 <= -0.05000000074505806f) {
		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(-0.05000000074505806))
		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(-0.05000000074505806))
		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 < -0.0500000007

   1. Initial program 53.9%

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

      1. +-commutative 53.9%

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

      2. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 15.0%

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

      2. div-sub 14.8%

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

      3. pow2 14.8%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 14.8%

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

      7. pow2 14.8%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 5.3%

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

      11. hypot-udef 5.3%

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

      12. hypot-udef 5.3%

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

      13. add-sqr-sqrt 5.3%

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

      14. metadata-eval 5.3%

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

      15. add-sqr-sqrt -0.0%

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

      16. fabs-sqr -0.0%

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

      17. add-sqr-sqrt 17.2%

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

   5. Applied egg-rr 17.2%

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

      1. unpow2 17.2%

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

      2. div-sub 19.1%

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

      3. unpow2 19.1%

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

      4. unpow2 19.1%

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

      5. unpow2 19.1%

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

      6. +-commutative 19.1%

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

      7. associate--r+ 50.8%

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

      8. +-inverses 99.9%

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

      9. metadata-eval 99.9%

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

      10. metadata-eval 99.9%

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

      11. associate-/r* 99.9%

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

      12. neg-mul-1 99.9%

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

      13. sub-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. distribute-neg-in 99.9%

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

      16. remove-double-neg 99.9%

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

      17. sub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -0.0500000007 < x < 0.0500000007

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

      2. hypot-1-def 19.6%

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

   3. Simplified 19.6%

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

      1. flip-+ 19.5%

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

      2. clear-num 19.5%

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

      3. add-sqr-sqrt 11.4%

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

      4. fabs-sqr 11.4%

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

      5. add-sqr-sqrt 19.7%

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

      6. pow2 19.7%

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

      7. add-sqr-sqrt 11.4%

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

      8. fabs-sqr 11.4%

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

      9. add-sqr-sqrt 19.7%

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

      10. pow2 19.7%

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

      11. hypot-udef 19.7%

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

      12. hypot-udef 19.6%

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

      13. add-sqr-sqrt 19.7%

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

      14. metadata-eval 19.7%

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

   5. Applied egg-rr 19.7%

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

   6. Taylor expanded in x around 0 100.0%

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

   if 0.0500000007 < x

   1. Initial program 50.8%

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

      1. +-commutative 50.8%

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

      2. hypot-1-def 98.8%

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

   3. Simplified 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. log-prod 98.8%

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

      3. metadata-eval 98.8%

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

      4. *-un-lft-identity 98.8%

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

      5. *-un-lft-identity 98.8%

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

      6. add-sqr-sqrt 98.7%

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

      7. fabs-sqr 98.7%

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

      8. add-sqr-sqrt 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. +-lft-identity 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 99.6%

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

Alternative 4: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.5)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 0.20000000298023224)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (+ x (/ 0.5 x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.5f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.5))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(x + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.5))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + (x + (single(0.5) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 49.3%

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

      1. +-commutative 49.3%

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

      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. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 98.9%

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

      1. associate--l+ 98.8%

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

      2. unpow1 98.8%

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

      3. sqr-pow -0.0%

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

      4. fabs-sqr -0.0%

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

      5. sqr-pow 10.8%

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

      6. unpow1 10.8%

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

      7. associate-+r- 97.5%

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

      8. mul-1-neg 97.5%

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

      9. sub-neg 97.5%

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

      10. +-inverses 97.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      11. neg-sub0 97.5%

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

      12. associate-*r/ 97.5%

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

      13. metadata-eval 97.5%

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

      14. distribute-neg-frac 97.5%

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

      15. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

      1. clear-num 97.5%

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

      2. log-div 97.5%

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

      3. metadata-eval 97.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{x}{-0.5}\right), x\right) \]

      4. div-inv 97.5%

         \[\leadsto \mathsf{copysign}\left(0 - \log \color{blue}{\left(x \cdot \frac{1}{-0.5}\right)}, x\right) \]

      5. metadata-eval 97.5%

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

   8. Applied egg-rr 97.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(x \cdot -2\right)}, x\right) \]
   9. Step-by-step derivation

      1. neg-sub0 97.5%

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

   10. Simplified 97.5%

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

   if -1.5 < x < 0.200000003

   1. Initial program 23.1%

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

      1. +-commutative 23.1%

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

      2. hypot-1-def 23.2%

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

   3. Simplified 23.2%

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

      1. flip-+ 22.9%

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

      2. clear-num 22.9%

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

      3. add-sqr-sqrt 11.6%

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

      4. fabs-sqr 11.6%

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

      5. add-sqr-sqrt 23.2%

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

      6. pow2 23.2%

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

      7. add-sqr-sqrt 11.6%

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

      8. fabs-sqr 11.6%

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

      9. add-sqr-sqrt 23.2%

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

      10. pow2 23.2%

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

      11. hypot-udef 23.2%

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

      12. hypot-udef 23.1%

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

      13. add-sqr-sqrt 23.4%

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

      14. metadata-eval 23.4%

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

   5. Applied egg-rr 23.4%

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

   6. Taylor expanded in x around 0 98.1%

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

   if 0.200000003 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around inf 98.8%

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

      1. associate-+r+ 98.8%

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

      2. +-commutative 98.8%

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

      3. +-commutative 98.8%

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

      4. unpow1 98.8%

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

      5. sqr-pow 98.7%

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

      6. fabs-sqr 98.7%

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

      7. sqr-pow 98.8%

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

      8. unpow1 98.8%

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

      9. associate-*r/ 98.8%

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

      10. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.5)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 0.20000000298023224)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.5f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.5))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.5))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 49.3%

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

      1. +-commutative 49.3%

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

      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. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 98.9%

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

      1. associate--l+ 98.8%

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

      2. unpow1 98.8%

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

      3. sqr-pow -0.0%

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

      4. fabs-sqr -0.0%

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

      5. sqr-pow 10.8%

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

      6. unpow1 10.8%

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

      7. associate-+r- 97.5%

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

      8. mul-1-neg 97.5%

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

      9. sub-neg 97.5%

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

      10. +-inverses 97.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      11. neg-sub0 97.5%

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

      12. associate-*r/ 97.5%

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

      13. metadata-eval 97.5%

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

      14. distribute-neg-frac 97.5%

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

      15. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

      1. clear-num 97.5%

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

      2. log-div 97.5%

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

      3. metadata-eval 97.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{x}{-0.5}\right), x\right) \]

      4. div-inv 97.5%

         \[\leadsto \mathsf{copysign}\left(0 - \log \color{blue}{\left(x \cdot \frac{1}{-0.5}\right)}, x\right) \]

      5. metadata-eval 97.5%

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

   8. Applied egg-rr 97.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(x \cdot -2\right)}, x\right) \]
   9. Step-by-step derivation

      1. neg-sub0 97.5%

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

   10. Simplified 97.5%

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

   if -1.5 < x < 0.200000003

   1. Initial program 23.1%

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

      1. +-commutative 23.1%

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

      2. hypot-1-def 23.2%

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

   3. Simplified 23.2%

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

      1. flip-+ 22.9%

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

      2. clear-num 22.9%

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

      3. add-sqr-sqrt 11.6%

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

      4. fabs-sqr 11.6%

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

      5. add-sqr-sqrt 23.2%

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

      6. pow2 23.2%

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

      7. add-sqr-sqrt 11.6%

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

      8. fabs-sqr 11.6%

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

      9. add-sqr-sqrt 23.2%

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

      10. pow2 23.2%

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

      11. hypot-udef 23.2%

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

      12. hypot-udef 23.1%

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

      13. add-sqr-sqrt 23.4%

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

      14. metadata-eval 23.4%

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

   5. Applied egg-rr 23.4%

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

   6. Taylor expanded in x around 0 98.1%

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

   if 0.200000003 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around inf 97.1%

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

      1. unpow1 97.1%

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

      2. sqr-pow 97.1%

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

      3. fabs-sqr 97.1%

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

      4. sqr-pow 97.1%

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

      5. unpow1 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 6: 83.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (- x)) x)
   (if (<= x 0.20000000298023224) (copysign x x) (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 48.3%

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

      1. +-commutative 48.3%

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

      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. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 44.6%

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

      1. mul-1-neg 44.6%

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

   6. Simplified 44.6%

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

   if -10 < x < 0.200000003

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. hypot-1-def 23.7%

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

   3. Simplified 23.7%

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

      1. flip-+ 23.4%

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

      2. clear-num 23.4%

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

      3. add-sqr-sqrt 11.5%

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

      4. fabs-sqr 11.5%

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

      5. add-sqr-sqrt 23.7%

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

      6. pow2 23.7%

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

      7. add-sqr-sqrt 11.5%

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

      8. fabs-sqr 11.5%

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

      9. add-sqr-sqrt 23.7%

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

      10. pow2 23.7%

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

      11. hypot-udef 23.8%

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

      12. hypot-udef 23.7%

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

      13. add-sqr-sqrt 23.9%

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

      14. metadata-eval 23.9%

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

   5. Applied egg-rr 23.9%

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

   6. Taylor expanded in x around 0 95.9%

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

   if 0.200000003 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around inf 97.1%

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

      1. unpow1 97.1%

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

      2. sqr-pow 97.1%

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

      3. fabs-sqr 97.1%

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

      4. sqr-pow 97.1%

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

      5. unpow1 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 7: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.5)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.20000000298023224) (copysign x x) (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.5f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.5))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.5))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 49.3%

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

      1. +-commutative 49.3%

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

      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. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 98.9%

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

      1. associate--l+ 98.8%

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

      2. unpow1 98.8%

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

      3. sqr-pow -0.0%

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

      4. fabs-sqr -0.0%

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

      5. sqr-pow 10.8%

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

      6. unpow1 10.8%

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

      7. associate-+r- 97.5%

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

      8. mul-1-neg 97.5%

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

      9. sub-neg 97.5%

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

      10. +-inverses 97.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      11. neg-sub0 97.5%

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

      12. associate-*r/ 97.5%

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

      13. metadata-eval 97.5%

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

      14. distribute-neg-frac 97.5%

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

      15. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

   if -1.5 < x < 0.200000003

   1. Initial program 23.1%

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

      1. +-commutative 23.1%

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

      2. hypot-1-def 23.2%

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

   3. Simplified 23.2%

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

      1. flip-+ 22.9%

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

      2. clear-num 22.9%

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

      3. add-sqr-sqrt 11.6%

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

      4. fabs-sqr 11.6%

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

      5. add-sqr-sqrt 23.2%

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

      6. pow2 23.2%

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

      7. add-sqr-sqrt 11.6%

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

      8. fabs-sqr 11.6%

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

      9. add-sqr-sqrt 23.2%

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

      10. pow2 23.2%

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

      11. hypot-udef 23.2%

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

      12. hypot-udef 23.1%

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

      13. add-sqr-sqrt 23.4%

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

      14. metadata-eval 23.4%

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

   5. Applied egg-rr 23.4%

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

   6. Taylor expanded in x around 0 96.4%

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

   if 0.200000003 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around inf 97.1%

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

      1. unpow1 97.1%

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

      2. sqr-pow 97.1%

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

      3. fabs-sqr 97.1%

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

      4. sqr-pow 97.1%

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

      5. unpow1 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 8: 97.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.5)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 0.20000000298023224) (copysign x x) (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.5f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.5))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.5))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 49.3%

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

      1. +-commutative 49.3%

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

      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. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 98.9%

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

      1. associate--l+ 98.8%

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

      2. unpow1 98.8%

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

      3. sqr-pow -0.0%

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

      4. fabs-sqr -0.0%

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

      5. sqr-pow 10.8%

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

      6. unpow1 10.8%

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

      7. associate-+r- 97.5%

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

      8. mul-1-neg 97.5%

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

      9. sub-neg 97.5%

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

      10. +-inverses 97.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      11. neg-sub0 97.5%

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

      12. associate-*r/ 97.5%

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

      13. metadata-eval 97.5%

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

      14. distribute-neg-frac 97.5%

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

      15. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

      1. clear-num 97.5%

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

      2. log-div 97.5%

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

      3. metadata-eval 97.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{x}{-0.5}\right), x\right) \]

      4. div-inv 97.5%

         \[\leadsto \mathsf{copysign}\left(0 - \log \color{blue}{\left(x \cdot \frac{1}{-0.5}\right)}, x\right) \]

      5. metadata-eval 97.5%

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

   8. Applied egg-rr 97.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(x \cdot -2\right)}, x\right) \]
   9. Step-by-step derivation

      1. neg-sub0 97.5%

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

   10. Simplified 97.5%

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

   if -1.5 < x < 0.200000003

   1. Initial program 23.1%

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

      1. +-commutative 23.1%

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

      2. hypot-1-def 23.2%

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

   3. Simplified 23.2%

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

      1. flip-+ 22.9%

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

      2. clear-num 22.9%

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

      3. add-sqr-sqrt 11.6%

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

      4. fabs-sqr 11.6%

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

      5. add-sqr-sqrt 23.2%

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

      6. pow2 23.2%

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

      7. add-sqr-sqrt 11.6%

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

      8. fabs-sqr 11.6%

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

      9. add-sqr-sqrt 23.2%

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

      10. pow2 23.2%

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

      11. hypot-udef 23.2%

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

      12. hypot-udef 23.1%

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

      13. add-sqr-sqrt 23.4%

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

      14. metadata-eval 23.4%

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

   5. Applied egg-rr 23.4%

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

   6. Taylor expanded in x around 0 96.4%

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

   if 0.200000003 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around inf 97.1%

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

      1. unpow1 97.1%

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

      2. sqr-pow 97.1%

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

      3. fabs-sqr 97.1%

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

      4. sqr-pow 97.1%

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

      5. unpow1 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 9: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0) (copysign (log (- x)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		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 < -1

   1. Initial program 51.3%

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

      1. +-commutative 51.3%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 43.3%

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

      1. mul-1-neg 43.3%

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

   6. Simplified 43.3%

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

   if -1 < x

   1. Initial program 32.2%

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

      1. +-commutative 32.2%

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

      2. hypot-1-def 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in x around 0 28.1%

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

      1. log1p-def 76.2%

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

      2. unpow1 76.2%

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

      3. sqr-pow 50.4%

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

      4. fabs-sqr 50.4%

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

      5. sqr-pow 76.1%

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

      6. unpow1 76.1%

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

   6. Simplified 76.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 10: 62.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.20000000298023224) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.200000003

   1. Initial program 30.1%

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

      1. +-commutative 30.1%

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

      2. hypot-1-def 43.5%

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

   3. Simplified 43.5%

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

      1. flip-+ 18.5%

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

      2. clear-num 18.5%

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

      3. add-sqr-sqrt 8.5%

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

      4. fabs-sqr 8.5%

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

      5. add-sqr-sqrt 19.9%

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

      6. pow2 19.9%

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

      7. add-sqr-sqrt 8.5%

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

      8. fabs-sqr 8.5%

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

      9. add-sqr-sqrt 19.9%

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

      10. pow2 19.9%

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

      11. hypot-udef 19.9%

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

      12. hypot-udef 19.8%

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

      13. add-sqr-sqrt 20.0%

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

      14. metadata-eval 20.0%

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

   5. Applied egg-rr 20.0%

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

   6. Taylor expanded in x around 0 73.8%

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

   if 0.200000003 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 44.7%

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

      1. log1p-def 44.7%

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

      2. unpow1 44.7%

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

      3. sqr-pow 44.7%

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

      4. fabs-sqr 44.7%

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

      5. sqr-pow 44.7%

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

      6. unpow1 44.7%

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

   6. Simplified 44.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 11: 53.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign x x))

C

float code(float x) {
	return copysignf(x, x);
}

Julia

function code(x)
	return copysign(x, x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(x);
end

Derivation

1. Initial program 35.9%

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

   1. +-commutative 35.9%

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

   2. hypot-1-def 59.7%

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

3. Simplified 59.7%

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

   1. flip-+ 15.0%

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

   2. clear-num 15.0%

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

   3. add-sqr-sqrt 8.1%

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

   4. fabs-sqr 8.1%

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

   5. add-sqr-sqrt 15.9%

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

   6. pow2 15.9%

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

   7. add-sqr-sqrt 8.0%

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

   8. fabs-sqr 8.0%

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

   9. add-sqr-sqrt 15.9%

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

   10. pow2 15.9%

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

   11. hypot-udef 15.9%

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

   12. hypot-udef 15.8%

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

   13. add-sqr-sqrt 15.9%

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

   14. metadata-eval 15.9%

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

5. Applied egg-rr 15.9%

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

6. Taylor expanded in x around 0 55.4%

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

7. Final simplification 55.4%

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

Developer target: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))