Rust f32::asinh

Percentage Accurate: 37.6% → 99.6%

Time: 8.7s

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: 37.6% 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: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05999999865889549:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05999999865889549)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.05999999865889549f) {
		tmp = copysignf(logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05999999865889549))
		tmp = copysign(log(Float32(hypot(Float32(1.0), x) - x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.05999999865889549))
		tmp = sign(x) * abs(log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0599999987

   1. Initial program 53.1%

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

      1. +-commutative 53.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 9.8%

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

      2. div-sub 10.1%

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

      3. pow2 10.1%

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

      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 10.1%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 3.6%

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

   6. Applied egg-rr 13.3%

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

      1. div-sub 14.6%

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

      2. remove-double-neg 14.6%

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

      3. distribute-frac-neg2 14.6%

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

      4. neg-mul-1 14.6%

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

      5. associate-/r* 14.6%

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

      6. distribute-neg-frac2 14.6%

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

      7. fma-undefine 14.2%

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

      8. unpow2 14.2%

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

      9. associate--r+ 49.9%

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

      10. sub-neg 49.9%

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

      11. +-inverses 99.9%

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

      12. metadata-eval 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. neg-sub0 99.9%

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

      16. associate--r- 99.9%

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

      17. neg-sub0 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

   8. Simplified 99.9%

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

      1. flip-- 9.9%

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

      2. +-commutative 9.9%

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

      3. div-inv 9.9%

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

      4. hypot-1-def 9.8%

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

      5. hypot-1-def 9.6%

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

      6. add-sqr-sqrt 9.9%

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

      7. pow2 9.9%

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

      8. pow2 9.9%

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

   10. Applied egg-rr 9.9%

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

       1. associate-*r/ 9.9%

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

       2. *-rgt-identity 9.9%

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

       3. associate--l+ 12.3%

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

       4. +-inverses 15.6%

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

       5. metadata-eval 15.6%

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

   12. Simplified 15.6%

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

       1. remove-double-div 15.6%

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

       2. +-commutative 15.6%

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

   14. Applied egg-rr 15.6%

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

       1. add-sqr-sqrt -0.0%

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

       2. sqrt-unprod 53.1%

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

       3. sqr-neg 53.1%

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

       4. sqrt-unprod 99.9%

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

       5. add-sqr-sqrt 99.9%

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

       6. sub-neg 99.9%

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

   16. Applied egg-rr 99.9%

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

   if -0.0599999987 < x < 0.0500000007

   1. Initial program 20.5%

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

      1. +-commutative 20.5%

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

      2. hypot-1-def 20.6%

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

   3. Simplified 20.6%

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

      1. flip-+ 20.5%

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

      2. div-sub 20.5%

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

      3. pow2 20.5%

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

      4. add-sqr-sqrt 9.7%

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

      5. fabs-sqr 9.7%

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

      6. add-sqr-sqrt 20.5%

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

      7. add-sqr-sqrt 9.7%

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

      8. fabs-sqr 9.7%

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

      9. add-sqr-sqrt 19.7%

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

   6. Applied egg-rr 20.5%

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

      1. div-sub 20.5%

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

      2. remove-double-neg 20.5%

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

      3. distribute-frac-neg2 20.5%

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

      4. neg-mul-1 20.5%

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

      5. associate-/r* 20.5%

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

      6. distribute-neg-frac2 20.5%

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

      7. fma-undefine 20.5%

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

      8. unpow2 20.5%

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

      9. associate--r+ 20.7%

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

      10. sub-neg 20.7%

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

      11. +-inverses 20.7%

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

      12. metadata-eval 20.7%

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

      13. metadata-eval 20.7%

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

      14. metadata-eval 20.7%

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

      15. neg-sub0 20.7%

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

      16. associate--r- 20.7%

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

      17. neg-sub0 20.7%

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

      18. +-commutative 20.7%

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

      19. sub-neg 20.7%

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

   8. Simplified 20.7%

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

   9. Taylor expanded in x around 0 99.8%

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

       1. distribute-rgt-in 99.9%

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

       2. *-lft-identity 99.9%

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

       3. associate-*l* 99.9%

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

       4. unpow2 99.9%

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

       5. unpow3 99.9%

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

   11. Simplified 99.9%

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

   if 0.0500000007 < x

   1. Initial program 57.1%

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

      1. +-commutative 57.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 57.1%

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

      1. rem-square-sqrt 57.1%

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

      2. fabs-sqr 57.1%

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

      3. metadata-eval 57.1%

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

      4. unpow2 57.1%

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

      5. hypot-undefine 99.9%

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

      6. rem-square-sqrt 99.9%

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

   7. Simplified 99.9%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<=
      (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)
      -9.999999747378752e-5)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= -9.999999747378752e-5f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else {
		tmp = copysignf(log1pf((x + (hypotf(1.0f, x) + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x) <= Float32(-9.999999747378752e-5))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(hypot(Float32(1.0), x) + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -9.99999975e-5

   1. Initial program 55.1%

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

      1. +-commutative 55.1%

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

      2. hypot-1-def 97.7%

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

   3. Simplified 97.7%

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

      1. flip-+ 16.0%

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

      2. div-sub 16.2%

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

      3. pow2 16.2%

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

      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 16.2%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 9.3%

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

   6. Applied egg-rr 19.2%

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

      1. div-sub 20.5%

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

      2. remove-double-neg 20.5%

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

      3. distribute-frac-neg2 20.5%

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

      4. neg-mul-1 20.5%

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

      5. associate-/r* 20.5%

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

      6. distribute-neg-frac2 20.5%

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

      7. fma-undefine 20.1%

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

      8. unpow2 20.1%

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

      9. associate--r+ 52.5%

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

      10. sub-neg 52.5%

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

      11. +-inverses 97.8%

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

      12. metadata-eval 97.8%

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

      13. metadata-eval 97.8%

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

      14. metadata-eval 97.8%

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

      15. neg-sub0 97.8%

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

      16. associate--r- 97.8%

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

      17. neg-sub0 97.8%

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

      18. +-commutative 97.8%

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

      19. sub-neg 97.8%

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

   8. Simplified 97.8%

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

   if -9.99999975e-5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 30.6%

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

      1. +-commutative 30.6%

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

      2. hypot-1-def 45.5%

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

   3. Simplified 45.5%

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

      1. flip-+ 13.5%

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

      2. div-sub 13.6%

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

      3. pow2 13.6%

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

      4. add-sqr-sqrt 9.4%

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

      5. fabs-sqr 9.4%

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

      6. add-sqr-sqrt 13.6%

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

      7. add-sqr-sqrt 9.2%

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

      8. fabs-sqr 9.2%

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

      9. add-sqr-sqrt 13.6%

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

   6. Applied egg-rr 13.4%

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

      1. div-sub 13.4%

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

      2. remove-double-neg 13.4%

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

      3. distribute-frac-neg2 13.4%

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

      4. neg-mul-1 13.4%

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

      5. associate-/r* 13.4%

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

      6. distribute-neg-frac2 13.4%

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

      7. fma-undefine 13.4%

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

      8. unpow2 13.4%

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

      9. associate--r+ 13.5%

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

      10. sub-neg 13.5%

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

      11. +-inverses 13.5%

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

      12. metadata-eval 13.5%

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

      13. metadata-eval 13.5%

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

      14. metadata-eval 13.5%

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

      15. neg-sub0 15.5%

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

      16. associate--r- 15.5%

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

      17. neg-sub0 15.5%

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

      18. +-commutative 15.5%

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

      19. sub-neg 15.5%

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

   8. Simplified 15.5%

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

      1. flip-- 14.3%

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

      2. +-commutative 14.3%

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

      3. div-inv 14.3%

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

      4. hypot-1-def 14.7%

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

      5. hypot-1-def 14.1%

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

      6. add-sqr-sqrt 15.2%

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

      7. pow2 15.2%

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

      8. pow2 15.2%

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

   10. Applied egg-rr 15.2%

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

       1. associate-*r/ 15.2%

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

       2. *-rgt-identity 15.2%

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

       3. associate--l+ 29.6%

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

       4. +-inverses 45.5%

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

       5. metadata-eval 45.5%

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

   12. Simplified 45.5%

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

       1. log1p-expm1-u 45.5%

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

       2. expm1-undefine 45.5%

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

       3. add-exp-log 45.5%

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

       4. remove-double-div 45.5%

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

   14. Applied egg-rr 45.5%

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

       1. associate--l+ 99.2%

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

   16. Simplified 99.2%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (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 <= -2.0f) {
		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(-2.0))
		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(-2.0))
		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 < -2

   1. Initial program 51.4%

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

      1. +-commutative 51.4%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 6.5%

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

      2. div-sub 6.7%

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

      3. pow2 6.7%

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

      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 6.7%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 2.1%

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

   6. Applied egg-rr 10.0%

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

      1. div-sub 11.3%

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

      2. remove-double-neg 11.3%

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

      3. distribute-frac-neg2 11.3%

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

      4. neg-mul-1 11.3%

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

      5. associate-/r* 11.3%

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

      6. distribute-neg-frac2 11.3%

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

      7. fma-undefine 11.0%

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

      8. unpow2 11.0%

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

      9. associate--r+ 47.9%

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

      10. sub-neg 47.9%

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

      11. +-inverses 99.9%

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

      12. metadata-eval 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. neg-sub0 99.9%

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

      16. associate--r- 99.9%

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

      17. neg-sub0 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 97.4%

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

       1. clear-num 97.4%

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

       2. log-rec 97.4%

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

       3. div-inv 97.4%

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

       4. metadata-eval 97.4%

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

   11. Applied egg-rr 97.4%

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

   if -2 < x < 0.0500000007

   1. Initial program 22.3%

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

      1. +-commutative 22.3%

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

      2. hypot-1-def 22.5%

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

   3. Simplified 22.5%

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

      1. flip-+ 22.3%

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

      2. div-sub 22.3%

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

      3. pow2 22.3%

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

      4. add-sqr-sqrt 9.5%

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

      5. fabs-sqr 9.5%

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

      6. add-sqr-sqrt 22.3%

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

      7. add-sqr-sqrt 9.5%

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

      8. fabs-sqr 9.5%

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

      9. add-sqr-sqrt 20.2%

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

   6. Applied egg-rr 22.3%

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

      1. div-sub 22.4%

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

      2. remove-double-neg 22.4%

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

      3. distribute-frac-neg2 22.4%

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

      4. neg-mul-1 22.4%

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

      5. associate-/r* 22.4%

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

      6. distribute-neg-frac2 22.4%

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

      7. fma-undefine 22.4%

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

      8. unpow2 22.4%

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

      9. associate--r+ 22.6%

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

      10. sub-neg 22.6%

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

      11. +-inverses 22.6%

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

      12. metadata-eval 22.6%

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

      13. metadata-eval 22.6%

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

      14. metadata-eval 22.6%

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

      15. neg-sub0 22.6%

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

      16. associate--r- 22.6%

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

      17. neg-sub0 22.6%

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

      18. +-commutative 22.6%

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

      19. sub-neg 22.6%

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

   8. Simplified 22.6%

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

   9. Taylor expanded in x around 0 99.0%

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

       1. distribute-rgt-in 99.1%

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

       2. *-lft-identity 99.1%

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

       3. associate-*l* 99.1%

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

       4. unpow2 99.1%

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

       5. unpow3 99.1%

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

   11. Simplified 99.1%

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

   if 0.0500000007 < x

   1. Initial program 57.1%

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

      1. +-commutative 57.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 57.1%

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

      1. rem-square-sqrt 57.1%

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

      2. fabs-sqr 57.1%

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

      3. metadata-eval 57.1%

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

      4. unpow2 57.1%

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

      5. hypot-undefine 99.9%

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

      6. rem-square-sqrt 99.9%

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

   7. Simplified 99.9%

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

Alternative 4: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 51.4%

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

      1. +-commutative 51.4%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 6.5%

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

      2. div-sub 6.7%

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

      3. pow2 6.7%

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

      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 6.7%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 2.1%

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

   6. Applied egg-rr 10.0%

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

      1. div-sub 11.3%

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

      2. remove-double-neg 11.3%

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

      3. distribute-frac-neg2 11.3%

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

      4. neg-mul-1 11.3%

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

      5. associate-/r* 11.3%

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

      6. distribute-neg-frac2 11.3%

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

      7. fma-undefine 11.0%

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

      8. unpow2 11.0%

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

      9. associate--r+ 47.9%

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

      10. sub-neg 47.9%

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

      11. +-inverses 99.9%

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

      12. metadata-eval 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. neg-sub0 99.9%

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

      16. associate--r- 99.9%

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

      17. neg-sub0 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 97.4%

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

       1. clear-num 97.4%

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

       2. log-rec 97.4%

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

       3. div-inv 97.4%

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

       4. metadata-eval 97.4%

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

   11. Applied egg-rr 97.4%

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

   if -2 < x < 0.5

   1. Initial program 23.0%

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

      1. +-commutative 23.0%

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

      2. hypot-1-def 23.1%

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

   3. Simplified 23.1%

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

      1. flip-+ 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. div-sub 22.9%

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

      3. pow2 22.9%

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

      4. add-sqr-sqrt 10.2%

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

      5. fabs-sqr 10.2%

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

      6. add-sqr-sqrt 22.9%

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

      7. add-sqr-sqrt 10.2%

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

      8. fabs-sqr 10.2%

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

      9. add-sqr-sqrt 20.8%

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

   6. Applied egg-rr 22.9%

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

      1. div-sub 23.0%

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

      2. remove-double-neg 23.0%

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

      3. distribute-frac-neg2 23.0%

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

      4. neg-mul-1 23.0%

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

      5. associate-/r* 23.0%

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

      6. distribute-neg-frac2 23.0%

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

      7. fma-undefine 23.0%

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

      8. unpow2 23.0%

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

      9. associate--r+ 23.1%

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

      10. sub-neg 23.1%

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

      11. +-inverses 23.1%

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

      12. metadata-eval 23.1%

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

      13. metadata-eval 23.1%

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

      14. metadata-eval 23.1%

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

      15. neg-sub0 23.1%

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

      16. associate--r- 23.1%

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

      17. neg-sub0 23.1%

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

      18. +-commutative 23.1%

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

      19. sub-neg 23.1%

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

   8. Simplified 23.1%

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

   9. Taylor expanded in x around 0 98.7%

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

       1. distribute-rgt-in 98.7%

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

       2. *-lft-identity 98.7%

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

       3. associate-*l* 98.7%

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

       4. unpow2 98.7%

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

       5. unpow3 98.7%

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

   11. Simplified 98.7%

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

   if 0.5 < x

   1. Initial program 56.4%

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

      1. +-commutative 56.4%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 6.5%

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

      2. div-sub 6.6%

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

      3. pow2 6.6%

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

      4. add-sqr-sqrt 5.8%

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

      5. fabs-sqr 5.8%

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

      6. add-sqr-sqrt 6.6%

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

      7. add-sqr-sqrt 5.4%

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

      8. fabs-sqr 5.4%

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

      9. add-sqr-sqrt 6.6%

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

   6. Applied egg-rr 6.1%

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

      1. div-sub 6.2%

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

      2. remove-double-neg 6.2%

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

      3. distribute-frac-neg2 6.2%

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

      4. neg-mul-1 6.2%

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

      5. associate-/r* 6.2%

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

      6. distribute-neg-frac2 6.2%

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

      7. fma-undefine 6.2%

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

      8. unpow2 6.2%

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

      9. associate--r+ 6.2%

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

      10. sub-neg 6.2%

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

      11. +-inverses 6.2%

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

      12. metadata-eval 6.2%

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

      13. metadata-eval 6.2%

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

      14. metadata-eval 6.2%

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

      15. neg-sub0 12.1%

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

      16. associate--r- 12.1%

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

      17. neg-sub0 12.1%

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

      18. +-commutative 12.1%

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

      19. sub-neg 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in x around -inf -0.0%

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

       1. frac-2neg -0.0%

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

       2. div-inv -0.0%

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

       3. metadata-eval -0.0%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 54.2%

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

       6. sqr-neg 54.2%

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

       7. sqrt-unprod 97.7%

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

       8. add-sqr-sqrt 97.7%

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

   11. Applied egg-rr 97.7%

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

       1. associate-*r/ 97.7%

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

       2. metadata-eval 97.7%

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

   13. Simplified 97.7%

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

Alternative 5: 97.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 51.4%

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

      1. +-commutative 51.4%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 6.5%

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

      2. div-sub 6.7%

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

      3. pow2 6.7%

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

      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 6.7%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 2.1%

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

   6. Applied egg-rr 10.0%

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

      1. div-sub 11.3%

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

      2. remove-double-neg 11.3%

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

      3. distribute-frac-neg2 11.3%

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

      4. neg-mul-1 11.3%

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

      5. associate-/r* 11.3%

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

      6. distribute-neg-frac2 11.3%

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

      7. fma-undefine 11.0%

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

      8. unpow2 11.0%

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

      9. associate--r+ 47.9%

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

      10. sub-neg 47.9%

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

      11. +-inverses 99.9%

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

      12. metadata-eval 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. neg-sub0 99.9%

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

      16. associate--r- 99.9%

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

      17. neg-sub0 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 97.4%

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

       1. clear-num 97.4%

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

       2. log-rec 97.4%

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

       3. div-inv 97.4%

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

       4. metadata-eval 97.4%

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

   11. Applied egg-rr 97.4%

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

   if -2 < x < 0.5

   1. Initial program 23.0%

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

      1. +-commutative 23.0%

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

      2. hypot-1-def 23.1%

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

   3. Simplified 23.1%

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

      1. flip-+ 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. div-sub 22.9%

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

      3. pow2 22.9%

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

      4. add-sqr-sqrt 10.2%

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

      5. fabs-sqr 10.2%

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

      6. add-sqr-sqrt 22.9%

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

      7. add-sqr-sqrt 10.2%

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

      8. fabs-sqr 10.2%

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

      9. add-sqr-sqrt 20.8%

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

   6. Applied egg-rr 22.9%

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

      1. div-sub 23.0%

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

      2. remove-double-neg 23.0%

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

      3. distribute-frac-neg2 23.0%

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

      4. neg-mul-1 23.0%

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

      5. associate-/r* 23.0%

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

      6. distribute-neg-frac2 23.0%

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

      7. fma-undefine 23.0%

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

      8. unpow2 23.0%

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

      9. associate--r+ 23.1%

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

      10. sub-neg 23.1%

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

      11. +-inverses 23.1%

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

      12. metadata-eval 23.1%

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

      13. metadata-eval 23.1%

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

      14. metadata-eval 23.1%

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

      15. neg-sub0 23.1%

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

      16. associate--r- 23.1%

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

      17. neg-sub0 23.1%

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

      18. +-commutative 23.1%

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

      19. sub-neg 23.1%

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

   8. Simplified 23.1%

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

   9. Taylor expanded in x around 0 97.0%

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

   if 0.5 < x

   1. Initial program 56.4%

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

      1. +-commutative 56.4%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 6.5%

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

      2. div-sub 6.6%

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

      3. pow2 6.6%

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

      4. add-sqr-sqrt 5.8%

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

      5. fabs-sqr 5.8%

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

      6. add-sqr-sqrt 6.6%

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

      7. add-sqr-sqrt 5.4%

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

      8. fabs-sqr 5.4%

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

      9. add-sqr-sqrt 6.6%

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

   6. Applied egg-rr 6.1%

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

      1. div-sub 6.2%

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

      2. remove-double-neg 6.2%

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

      3. distribute-frac-neg2 6.2%

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

      4. neg-mul-1 6.2%

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

      5. associate-/r* 6.2%

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

      6. distribute-neg-frac2 6.2%

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

      7. fma-undefine 6.2%

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

      8. unpow2 6.2%

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

      9. associate--r+ 6.2%

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

      10. sub-neg 6.2%

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

      11. +-inverses 6.2%

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

      12. metadata-eval 6.2%

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

      13. metadata-eval 6.2%

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

      14. metadata-eval 6.2%

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

      15. neg-sub0 12.1%

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

      16. associate--r- 12.1%

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

      17. neg-sub0 12.1%

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

      18. +-commutative 12.1%

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

      19. sub-neg 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in x around -inf -0.0%

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

       1. frac-2neg -0.0%

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

       2. div-inv -0.0%

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

       3. metadata-eval -0.0%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 54.2%

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

       6. sqr-neg 54.2%

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

       7. sqrt-unprod 97.7%

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

       8. add-sqr-sqrt 97.7%

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

   11. Applied egg-rr 97.7%

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

       1. associate-*r/ 97.7%

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

       2. metadata-eval 97.7%

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

   13. Simplified 97.7%

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

Alternative 6: 97.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 51.4%

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

      1. +-commutative 51.4%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 6.5%

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

      2. div-sub 6.7%

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

      3. pow2 6.7%

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

      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 6.7%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 2.1%

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

   6. Applied egg-rr 10.0%

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

      1. div-sub 11.3%

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

      2. remove-double-neg 11.3%

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

      3. distribute-frac-neg2 11.3%

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

      4. neg-mul-1 11.3%

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

      5. associate-/r* 11.3%

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

      6. distribute-neg-frac2 11.3%

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

      7. fma-undefine 11.0%

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

      8. unpow2 11.0%

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

      9. associate--r+ 47.9%

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

      10. sub-neg 47.9%

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

      11. +-inverses 99.9%

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

      12. metadata-eval 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. neg-sub0 99.9%

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

      16. associate--r- 99.9%

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

      17. neg-sub0 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 97.4%

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

   if -2 < x < 0.5

   1. Initial program 23.0%

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

      1. +-commutative 23.0%

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

      2. hypot-1-def 23.1%

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

   3. Simplified 23.1%

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

      1. flip-+ 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. div-sub 22.9%

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

      3. pow2 22.9%

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

      4. add-sqr-sqrt 10.2%

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

      5. fabs-sqr 10.2%

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

      6. add-sqr-sqrt 22.9%

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

      7. add-sqr-sqrt 10.2%

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

      8. fabs-sqr 10.2%

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

      9. add-sqr-sqrt 20.8%

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

   6. Applied egg-rr 22.9%

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

      1. div-sub 23.0%

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

      2. remove-double-neg 23.0%

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

      3. distribute-frac-neg2 23.0%

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

      4. neg-mul-1 23.0%

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

      5. associate-/r* 23.0%

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

      6. distribute-neg-frac2 23.0%

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

      7. fma-undefine 23.0%

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

      8. unpow2 23.0%

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

      9. associate--r+ 23.1%

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

      10. sub-neg 23.1%

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

      11. +-inverses 23.1%

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

      12. metadata-eval 23.1%

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

      13. metadata-eval 23.1%

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

      14. metadata-eval 23.1%

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

      15. neg-sub0 23.1%

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

      16. associate--r- 23.1%

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

      17. neg-sub0 23.1%

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

      18. +-commutative 23.1%

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

      19. sub-neg 23.1%

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

   8. Simplified 23.1%

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

   9. Taylor expanded in x around 0 97.0%

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

   if 0.5 < x

   1. Initial program 56.4%

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

      1. +-commutative 56.4%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 6.5%

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

      2. div-sub 6.6%

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

      3. pow2 6.6%

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

      4. add-sqr-sqrt 5.8%

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

      5. fabs-sqr 5.8%

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

      6. add-sqr-sqrt 6.6%

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

      7. add-sqr-sqrt 5.4%

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

      8. fabs-sqr 5.4%

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

      9. add-sqr-sqrt 6.6%

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

   6. Applied egg-rr 6.1%

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

      1. div-sub 6.2%

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

      2. remove-double-neg 6.2%

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

      3. distribute-frac-neg2 6.2%

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

      4. neg-mul-1 6.2%

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

      5. associate-/r* 6.2%

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

      6. distribute-neg-frac2 6.2%

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

      7. fma-undefine 6.2%

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

      8. unpow2 6.2%

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

      9. associate--r+ 6.2%

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

      10. sub-neg 6.2%

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

      11. +-inverses 6.2%

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

      12. metadata-eval 6.2%

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

      13. metadata-eval 6.2%

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

      14. metadata-eval 6.2%

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

      15. neg-sub0 12.1%

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

      16. associate--r- 12.1%

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

      17. neg-sub0 12.1%

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

      18. +-commutative 12.1%

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

      19. sub-neg 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in x around -inf -0.0%

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

       1. frac-2neg -0.0%

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

       2. div-inv -0.0%

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

       3. metadata-eval -0.0%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 54.2%

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

       6. sqr-neg 54.2%

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

       7. sqrt-unprod 97.7%

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

       8. add-sqr-sqrt 97.7%

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

   11. Applied egg-rr 97.7%

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

       1. associate-*r/ 97.7%

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

       2. metadata-eval 97.7%

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

   13. Simplified 97.7%

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

Alternative 7: 97.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 51.4%

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

      1. +-commutative 51.4%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 6.5%

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

      2. div-sub 6.7%

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

      3. pow2 6.7%

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

      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 6.7%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 2.1%

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

   6. Applied egg-rr 10.0%

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

      1. div-sub 11.3%

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

      2. remove-double-neg 11.3%

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

      3. distribute-frac-neg2 11.3%

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

      4. neg-mul-1 11.3%

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

      5. associate-/r* 11.3%

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

      6. distribute-neg-frac2 11.3%

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

      7. fma-undefine 11.0%

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

      8. unpow2 11.0%

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

      9. associate--r+ 47.9%

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

      10. sub-neg 47.9%

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

      11. +-inverses 99.9%

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

      12. metadata-eval 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. neg-sub0 99.9%

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

      16. associate--r- 99.9%

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

      17. neg-sub0 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 97.4%

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

   if -2 < x < 0.5

   1. Initial program 23.0%

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

      1. +-commutative 23.0%

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

      2. hypot-1-def 23.1%

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

   3. Simplified 23.1%

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

      1. flip-+ 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. div-sub 22.9%

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

      3. pow2 22.9%

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

      4. add-sqr-sqrt 10.2%

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

      5. fabs-sqr 10.2%

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

      6. add-sqr-sqrt 22.9%

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

      7. add-sqr-sqrt 10.2%

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

      8. fabs-sqr 10.2%

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

      9. add-sqr-sqrt 20.8%

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

   6. Applied egg-rr 22.9%

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

      1. div-sub 23.0%

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

      2. remove-double-neg 23.0%

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

      3. distribute-frac-neg2 23.0%

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

      4. neg-mul-1 23.0%

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

      5. associate-/r* 23.0%

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

      6. distribute-neg-frac2 23.0%

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

      7. fma-undefine 23.0%

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

      8. unpow2 23.0%

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

      9. associate--r+ 23.1%

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

      10. sub-neg 23.1%

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

      11. +-inverses 23.1%

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

      12. metadata-eval 23.1%

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

      13. metadata-eval 23.1%

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

      14. metadata-eval 23.1%

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

      15. neg-sub0 23.1%

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

      16. associate--r- 23.1%

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

      17. neg-sub0 23.1%

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

      18. +-commutative 23.1%

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

      19. sub-neg 23.1%

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

   8. Simplified 23.1%

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

   9. Taylor expanded in x around 0 97.0%

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

   if 0.5 < x

   1. Initial program 56.4%

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

      1. +-commutative 56.4%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 97.7%

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

      1. +-commutative 97.7%

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

      2. rem-square-sqrt 97.7%

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

      3. fabs-sqr 97.7%

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

      4. rem-square-sqrt 97.7%

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

      5. *-inverses 97.7%

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

      6. metadata-eval 97.7%

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

   7. Simplified 97.7%

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

Alternative 8: 83.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -3

   1. Initial program 50.7%

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

      1. +-commutative 50.7%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around -inf 44.3%

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

      1. mul-1-neg 44.3%

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

   7. Simplified 44.3%

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

   if -3 < x < 0.5

   1. Initial program 23.6%

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

      1. +-commutative 23.6%

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

      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. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ 23.5%

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

      2. div-sub 23.5%

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

      3. pow2 23.5%

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

      4. add-sqr-sqrt 10.2%

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

      5. fabs-sqr 10.2%

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

      6. add-sqr-sqrt 23.5%

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

      7. add-sqr-sqrt 10.1%

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

      8. fabs-sqr 10.1%

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

      9. add-sqr-sqrt 20.9%

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

   6. Applied egg-rr 23.5%

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

      1. div-sub 23.6%

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

      2. remove-double-neg 23.6%

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

      3. distribute-frac-neg2 23.6%

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

      4. neg-mul-1 23.6%

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

      5. associate-/r* 23.6%

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

      6. distribute-neg-frac2 23.6%

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

      7. fma-undefine 23.6%

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

      8. unpow2 23.6%

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

      9. associate--r+ 23.7%

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

      10. sub-neg 23.7%

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

      11. +-inverses 23.7%

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

      12. metadata-eval 23.7%

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

      13. metadata-eval 23.7%

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

      14. metadata-eval 23.7%

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

      15. neg-sub0 23.7%

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

      16. associate--r- 23.7%

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

      17. neg-sub0 23.7%

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

      18. +-commutative 23.7%

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

      19. sub-neg 23.7%

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

   8. Simplified 23.7%

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

   9. Taylor expanded in x around 0 96.4%

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

   if 0.5 < x

   1. Initial program 56.4%

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

      1. +-commutative 56.4%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 97.7%

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

      1. +-commutative 97.7%

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

      2. rem-square-sqrt 97.7%

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

      3. fabs-sqr 97.7%

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

      4. rem-square-sqrt 97.7%

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

      5. *-inverses 97.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\color{blue}{1} + 1\right)\right), x\right) \]

      6. metadata-eval 97.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \color{blue}{2}\right), x\right) \]

   7. Simplified 97.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.9% 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 52.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. +-commutative 52.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]

      2. hypot-1-def 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 43.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 43.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   7. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   if -1 < x

   1. Initial program 33.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. +-commutative 33.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]

      2. hypot-1-def 47.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 27.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define 77.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

      2. rem-square-sqrt 46.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

      3. fabs-sqr 46.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]

      4. rem-square-sqrt 77.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right), x\right) \]

   7. Simplified 77.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.5) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 33.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. +-commutative 33.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]

      2. hypot-1-def 52.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Simplified 52.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ 16.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

      2. div-sub 16.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right|}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

      3. pow2 16.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      4. add-sqr-sqrt 6.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. fabs-sqr 6.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. add-sqr-sqrt 16.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{\color{blue}{x}}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. add-sqr-sqrt 6.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{x}^{2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. fabs-sqr 6.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{x}^{2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt 13.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{x}^{2}}{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   6. Applied egg-rr 18.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. div-sub 18.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

      2. remove-double-neg 18.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\left(-\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right)}, x\right) \]

      3. distribute-frac-neg2 18.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(-\color{blue}{\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

      4. neg-mul-1 18.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(-\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{\color{blue}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

      5. associate-/r* 18.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(-\color{blue}{\frac{\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-1}}{x - \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

      6. distribute-neg-frac2 18.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]

      7. fma-undefine 18.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      8. unpow2 18.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      9. associate--r+ 32.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      10. sub-neg 32.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{\color{blue}{\left({x}^{2} - {x}^{2}\right) + \left(-1\right)}}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      11. +-inverses 52.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{\color{blue}{0} + \left(-1\right)}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      12. metadata-eval 52.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{0 + \color{blue}{-1}}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      13. metadata-eval 52.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{\color{blue}{-1}}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      14. metadata-eval 52.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      15. neg-sub0 52.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

      16. associate--r- 52.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

      17. neg-sub0 52.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      18. +-commutative 52.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right), x\right) \]

      19. sub-neg 52.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

   8. Simplified 52.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]

   9. Taylor expanded in x around 0 64.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.5 < x

   1. Initial program 56.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. +-commutative 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]

      2. hypot-1-def 99.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 44.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. Step-by-step derivation

      1. log1p-define 44.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

      2. rem-square-sqrt 44.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

      3. fabs-sqr 44.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]

      4. rem-square-sqrt 44.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right), x\right) \]

   7. Simplified 44.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 54.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign x x))

C

float code(float x) {
	return copysignf(x, x);
}

Julia

function code(x)
	return copysign(x, x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(x);
end

Derivation

1. Initial program 38.9%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Step-by-step derivation

   1. +-commutative 38.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]

   2. hypot-1-def 63.0%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

3. Simplified 63.0%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. flip-+ 14.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   2. div-sub 14.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right|}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   3. pow2 14.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   4. add-sqr-sqrt 6.2%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   5. fabs-sqr 6.2%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   6. add-sqr-sqrt 14.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{\color{blue}{x}}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   7. add-sqr-sqrt 6.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{x}^{2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   8. fabs-sqr 6.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{x}^{2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   9. add-sqr-sqrt 12.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{x}^{2}}{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

6. Applied egg-rr 15.3%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
7. Step-by-step derivation

   1. div-sub 15.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   2. remove-double-neg 15.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\left(-\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right)}, x\right) \]

   3. distribute-frac-neg2 15.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(-\color{blue}{\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

   4. neg-mul-1 15.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(-\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{\color{blue}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

   5. associate-/r* 15.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(-\color{blue}{\frac{\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-1}}{x - \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

   6. distribute-neg-frac2 15.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]

   7. fma-undefine 15.7%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   8. unpow2 15.7%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{{x}^{2} - \left(\color{blue}{{x}^{2}} + 1\right)}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   9. associate--r+ 26.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   10. sub-neg 26.6%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{\color{blue}{\left({x}^{2} - {x}^{2}\right) + \left(-1\right)}}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   11. +-inverses 41.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{\color{blue}{0} + \left(-1\right)}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   12. metadata-eval 41.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{0 + \color{blue}{-1}}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   13. metadata-eval 41.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\frac{\color{blue}{-1}}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   14. metadata-eval 41.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   15. neg-sub0 43.1%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

   16. associate--r- 43.1%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

   17. neg-sub0 43.1%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   18. +-commutative 43.1%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right), x\right) \]

   19. sub-neg 43.1%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

8. Simplified 43.1%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]

9. Taylor expanded in x around 0 52.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
10. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

float code(float x) {
	float t_0 = 1.0f / fabsf(x);
	return copysignf(log1pf((fabsf(x) + (fabsf(x) / (hypotf(1.0f, t_0) + t_0)))), x);
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) / abs(x))
	return copysign(log1p(Float32(abs(x) + Float32(abs(x) / Float32(hypot(Float32(1.0), t_0) + t_0)))), x)
end

Reproduce

herbie shell --seed 2024180 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (! :herbie-platform default (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))