Rust f32::asinh

Percentage Accurate: 38.2% → 98.6%

Time: 6.5s

Alternatives: 12

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 38.2% 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: 98.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05000000074505806)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (+ (log x) (log 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.05000000074505806f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf((logf(x) + logf(2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05000000074505806))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(Float32(log(x) + log(Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.05000000074505806))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs((log(x) + log(single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 58.6%

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

      1. flip-+ 9.3%

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

      2. sqr-abs 9.3%

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

      3. add-sqr-sqrt 8.9%

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

      4. div-sub 8.8%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 3.3%

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

      8. +-commutative 3.3%

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

      9. hypot-1-def 3.3%

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

      10. fma-def 3.3%

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

      11. add-sqr-sqrt -0.0%

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

      12. fabs-sqr -0.0%

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

      13. add-sqr-sqrt 12.2%

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

      14. +-commutative 12.2%

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

      15. hypot-1-def 12.3%

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

   3. Applied egg-rr 12.3%

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

      1. div-sub 13.4%

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

      2. *-lft-identity 13.4%

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

      3. metadata-eval 13.4%

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

      4. times-frac 13.4%

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

      5. fma-udef 13.4%

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

      6. associate--r+ 55.9%

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

      7. +-inverses 98.4%

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

      8. metadata-eval 98.4%

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

      9. metadata-eval 98.4%

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

      10. neg-mul-1 98.4%

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

      11. neg-sub0 98.4%

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

      12. associate--r- 98.4%

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

      13. neg-sub0 98.4%

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

      14. +-commutative 98.4%

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

      15. sub-neg 98.4%

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

   5. Simplified 98.4%

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

   if -0.0500000007 < x < 1

   1. Initial program 23.2%

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

      1. flip-+ 23.3%

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

      2. sqr-abs 23.3%

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

      3. add-sqr-sqrt 23.2%

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

      4. div-sub 23.3%

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

      5. add-sqr-sqrt 12.7%

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

      6. fabs-sqr 12.7%

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

      7. add-sqr-sqrt 22.6%

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

      8. +-commutative 22.6%

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

      9. hypot-1-def 22.6%

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

      10. fma-def 22.6%

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

      11. add-sqr-sqrt 12.7%

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

      12. fabs-sqr 12.7%

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

      13. add-sqr-sqrt 23.2%

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

      14. +-commutative 23.2%

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

      15. hypot-1-def 23.1%

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

   3. Applied egg-rr 23.1%

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

      1. div-sub 23.1%

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

      2. *-lft-identity 23.1%

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

      3. metadata-eval 23.1%

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

      4. times-frac 23.1%

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

      5. fma-udef 23.1%

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

      6. associate--r+ 23.2%

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

      7. +-inverses 23.2%

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

      8. metadata-eval 23.2%

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

      9. metadata-eval 23.2%

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

      10. neg-mul-1 23.2%

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

      11. neg-sub0 23.2%

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

      12. associate--r- 23.2%

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

      13. neg-sub0 23.2%

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

      14. +-commutative 23.2%

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

      15. sub-neg 23.2%

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

   5. Simplified 23.2%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1 < x

   1. Initial program 53.9%

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

      1. flip-+ 6.3%

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

      2. sqr-abs 6.3%

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

      3. add-sqr-sqrt 6.2%

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

      4. div-sub 6.3%

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

      5. add-sqr-sqrt 5.4%

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

      6. fabs-sqr 5.4%

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

      7. add-sqr-sqrt 6.3%

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

      8. +-commutative 6.3%

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

      9. hypot-1-def 5.5%

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

      10. fma-def 5.5%

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

      11. add-sqr-sqrt 6.0%

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

      12. fabs-sqr 6.0%

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

      13. add-sqr-sqrt 5.5%

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

      14. +-commutative 5.5%

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

      15. hypot-1-def 6.3%

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

   3. Applied egg-rr 6.3%

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

      1. div-sub 6.3%

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

      2. *-lft-identity 6.3%

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

      3. metadata-eval 6.3%

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

      4. times-frac 6.3%

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

      5. fma-udef 6.3%

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

      6. associate--r+ 6.3%

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

      7. +-inverses 6.3%

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

      8. metadata-eval 6.3%

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

      9. metadata-eval 6.3%

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

      10. neg-mul-1 6.3%

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

      11. neg-sub0 12.1%

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

      12. associate--r- 12.1%

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

      13. neg-sub0 12.1%

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

      14. +-commutative 12.1%

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

      15. sub-neg 12.1%

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

   5. Simplified 12.1%

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

   6. Taylor expanded in x around inf 97.3%

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

      1. +-commutative 97.3%

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

      2. mul-1-neg 97.3%

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

      3. log-rec 97.3%

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

      4. remove-double-neg 97.3%

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

   8. Simplified 97.3%

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

4. Final simplification 98.5%

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

Alternative 2: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.0005000000237487257:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<=
      (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)
      -0.0005000000237487257)
   (copysign (- (log (- (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) <= -0.0005000000237487257f) {
		tmp = copysignf(-logf((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(-0.0005000000237487257))
		tmp = copysign(Float32(-log(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) 1)))) x) < -5.00000024e-4

   1. Initial program 60.2%

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

      1. flip-+ 13.6%

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

      2. frac-2neg 13.6%

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

      3. log-div 13.5%

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

      4. sqr-abs 13.5%

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

      5. add-sqr-sqrt 13.3%

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

      6. fma-def 13.3%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 17.3%

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

      10. +-commutative 17.3%

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

   3. Applied egg-rr 17.2%

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

      1. neg-mul-1 17.2%

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

      2. fma-udef 17.2%

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

      3. associate--r+ 57.5%

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

      4. +-inverses 97.6%

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

      5. metadata-eval 97.6%

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

      6. metadata-eval 97.6%

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

      7. metadata-eval 97.6%

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

      8. neg-sub0 97.6%

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

      9. neg-sub0 97.6%

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

      10. associate--r- 97.6%

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

      11. neg-sub0 97.6%

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

      12. +-commutative 97.6%

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

      13. sub-neg 97.6%

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

   5. Simplified 97.6%

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

   if -5.00000024e-4 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 33.2%

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

      1. *-un-lft-identity 33.2%

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

      2. *-commutative 33.2%

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

      3. log-prod 33.2%

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

      4. add-sqr-sqrt 28.1%

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

      5. fabs-sqr 28.1%

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

      6. add-sqr-sqrt 33.4%

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

      7. +-commutative 33.4%

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

      8. hypot-1-def 49.0%

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

      9. metadata-eval 49.0%

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

   3. Applied egg-rr 49.0%

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

      1. +-rgt-identity 49.0%

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

   5. Simplified 49.0%

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

      1. log1p-expm1-u 49.0%

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

      2. expm1-udef 49.0%

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

      3. add-exp-log 49.0%

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

   7. Applied egg-rr 49.0%

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

      1. associate--l+ 96.7%

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

   9. Simplified 96.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.0005000000237487257:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 3: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 - \frac{0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \log 2, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ 1.0 (- (* x -2.0) (/ 0.5 x)))) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (+ (log x) (log 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((1.0f / ((x * -2.0f) - (0.5f / x)))), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf((logf(x) + logf(2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(Float32(x * Float32(-2.0)) - Float32(Float32(0.5) / x)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(Float32(log(x) + log(Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(1.0) / ((x * single(-2.0)) - (single(0.5) / x)))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs((log(x) + log(single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.8%

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

      1. flip-+ 5.3%

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

      2. sqr-abs 5.3%

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

      3. add-sqr-sqrt 4.7%

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

      4. div-sub 4.8%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 1.7%

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

      8. +-commutative 1.7%

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

      9. hypot-1-def 1.7%

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

      10. fma-def 1.7%

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

      11. add-sqr-sqrt -0.0%

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

      12. fabs-sqr -0.0%

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

      13. add-sqr-sqrt 8.1%

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

      14. +-commutative 8.1%

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

      15. hypot-1-def 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. div-sub 9.4%

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

      2. *-lft-identity 9.4%

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

      3. metadata-eval 9.4%

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

      4. times-frac 9.4%

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

      5. fma-udef 9.4%

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

      6. associate--r+ 53.9%

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

      7. +-inverses 98.5%

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

      8. metadata-eval 98.5%

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

      9. metadata-eval 98.5%

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

      10. neg-mul-1 98.5%

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

      11. neg-sub0 98.5%

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

      12. associate--r- 98.5%

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

      13. neg-sub0 98.5%

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

      14. +-commutative 98.5%

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

      15. sub-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around -inf 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*r/ 98.4%

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

      3. metadata-eval 98.4%

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

   8. Simplified 98.4%

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

   if -10 < x < 1

   1. Initial program 24.9%

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

      1. flip-+ 25.0%

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

      2. sqr-abs 25.0%

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

      3. add-sqr-sqrt 24.9%

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

      4. div-sub 25.0%

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

      5. add-sqr-sqrt 12.4%

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

      6. fabs-sqr 12.4%

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

      7. add-sqr-sqrt 23.0%

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

      8. +-commutative 23.0%

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

      9. hypot-1-def 23.0%

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

      10. fma-def 22.9%

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

      11. add-sqr-sqrt 12.4%

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

      12. fabs-sqr 12.4%

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

      13. add-sqr-sqrt 25.0%

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

      14. +-commutative 25.0%

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

      15. hypot-1-def 24.9%

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

   3. Applied egg-rr 24.9%

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

      1. div-sub 24.9%

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

      2. *-lft-identity 24.9%

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

      3. metadata-eval 24.9%

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

      4. times-frac 24.9%

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

      5. fma-udef 24.9%

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

      6. associate--r+ 25.0%

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

      7. +-inverses 25.0%

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

      8. metadata-eval 25.0%

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

      9. metadata-eval 25.0%

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

      10. neg-mul-1 25.0%

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

      11. neg-sub0 25.0%

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

      12. associate--r- 25.0%

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

      13. neg-sub0 25.0%

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

      14. +-commutative 25.0%

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

      15. sub-neg 25.0%

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

   5. Simplified 25.0%

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

   6. Taylor expanded in x around 0 98.3%

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

   if 1 < x

   1. Initial program 53.9%

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

      1. flip-+ 6.3%

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

      2. sqr-abs 6.3%

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

      3. add-sqr-sqrt 6.2%

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

      4. div-sub 6.3%

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

      5. add-sqr-sqrt 5.4%

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

      6. fabs-sqr 5.4%

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

      7. add-sqr-sqrt 6.3%

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

      8. +-commutative 6.3%

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

      9. hypot-1-def 5.5%

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

      10. fma-def 5.5%

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

      11. add-sqr-sqrt 6.0%

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

      12. fabs-sqr 6.0%

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

      13. add-sqr-sqrt 5.5%

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

      14. +-commutative 5.5%

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

      15. hypot-1-def 6.3%

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

   3. Applied egg-rr 6.3%

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

      1. div-sub 6.3%

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

      2. *-lft-identity 6.3%

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

      3. metadata-eval 6.3%

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

      4. times-frac 6.3%

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

      5. fma-udef 6.3%

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

      6. associate--r+ 6.3%

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

      7. +-inverses 6.3%

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

      8. metadata-eval 6.3%

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

      9. metadata-eval 6.3%

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

      10. neg-mul-1 6.3%

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

      11. neg-sub0 12.1%

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

      12. associate--r- 12.1%

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

      13. neg-sub0 12.1%

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

      14. +-commutative 12.1%

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

      15. sub-neg 12.1%

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

   5. Simplified 12.1%

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

   6. Taylor expanded in x around inf 97.3%

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

      1. +-commutative 97.3%

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

      2. mul-1-neg 97.3%

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

      3. log-rec 97.3%

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

      4. remove-double-neg 97.3%

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

   8. Simplified 97.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 - \frac{0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \log 2, x\right)\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05000000074505806)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (+ (log x) (log 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.05000000074505806f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf((logf(x) + logf(2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05000000074505806))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(Float32(log(x) + log(Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.05000000074505806))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs((log(x) + log(single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 58.6%

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

      1. flip-+ 9.3%

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

      2. frac-2neg 9.3%

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

      3. log-div 9.4%

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

      4. sqr-abs 9.4%

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

      5. add-sqr-sqrt 8.9%

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

      6. fma-def 8.9%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 13.3%

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

      10. +-commutative 13.3%

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

   3. Applied egg-rr 13.4%

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

      1. neg-mul-1 13.4%

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

      2. fma-udef 13.4%

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

      3. associate--r+ 55.8%

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

      4. +-inverses 98.3%

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

      5. metadata-eval 98.3%

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

      6. metadata-eval 98.3%

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

      7. metadata-eval 98.3%

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

      8. neg-sub0 98.3%

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

      9. neg-sub0 98.3%

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

      10. associate--r- 98.3%

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

      11. neg-sub0 98.3%

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

      12. +-commutative 98.3%

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

      13. sub-neg 98.3%

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

   5. Simplified 98.3%

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

   if -0.0500000007 < x < 1

   1. Initial program 23.2%

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

      1. flip-+ 23.3%

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

      2. sqr-abs 23.3%

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

      3. add-sqr-sqrt 23.2%

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

      4. div-sub 23.3%

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

      5. add-sqr-sqrt 12.7%

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

      6. fabs-sqr 12.7%

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

      7. add-sqr-sqrt 22.6%

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

      8. +-commutative 22.6%

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

      9. hypot-1-def 22.6%

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

      10. fma-def 22.6%

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

      11. add-sqr-sqrt 12.7%

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

      12. fabs-sqr 12.7%

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

      13. add-sqr-sqrt 23.2%

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

      14. +-commutative 23.2%

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

      15. hypot-1-def 23.1%

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

   3. Applied egg-rr 23.1%

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

      1. div-sub 23.1%

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

      2. *-lft-identity 23.1%

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

      3. metadata-eval 23.1%

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

      4. times-frac 23.1%

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

      5. fma-udef 23.1%

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

      6. associate--r+ 23.2%

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

      7. +-inverses 23.2%

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

      8. metadata-eval 23.2%

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

      9. metadata-eval 23.2%

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

      10. neg-mul-1 23.2%

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

      11. neg-sub0 23.2%

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

      12. associate--r- 23.2%

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

      13. neg-sub0 23.2%

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

      14. +-commutative 23.2%

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

      15. sub-neg 23.2%

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

   5. Simplified 23.2%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1 < x

   1. Initial program 53.9%

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

      1. flip-+ 6.3%

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

      2. sqr-abs 6.3%

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

      3. add-sqr-sqrt 6.2%

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

      4. div-sub 6.3%

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

      5. add-sqr-sqrt 5.4%

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

      6. fabs-sqr 5.4%

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

      7. add-sqr-sqrt 6.3%

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

      8. +-commutative 6.3%

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

      9. hypot-1-def 5.5%

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

      10. fma-def 5.5%

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

      11. add-sqr-sqrt 6.0%

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

      12. fabs-sqr 6.0%

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

      13. add-sqr-sqrt 5.5%

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

      14. +-commutative 5.5%

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

      15. hypot-1-def 6.3%

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

   3. Applied egg-rr 6.3%

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

      1. div-sub 6.3%

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

      2. *-lft-identity 6.3%

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

      3. metadata-eval 6.3%

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

      4. times-frac 6.3%

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

      5. fma-udef 6.3%

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

      6. associate--r+ 6.3%

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

      7. +-inverses 6.3%

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

      8. metadata-eval 6.3%

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

      9. metadata-eval 6.3%

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

      10. neg-mul-1 6.3%

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

      11. neg-sub0 12.1%

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

      12. associate--r- 12.1%

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

      13. neg-sub0 12.1%

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

      14. +-commutative 12.1%

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

      15. sub-neg 12.1%

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

   5. Simplified 12.1%

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

   6. Taylor expanded in x around inf 97.3%

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

      1. +-commutative 97.3%

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

      2. mul-1-neg 97.3%

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

      3. log-rec 97.3%

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

      4. remove-double-neg 97.3%

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

   8. Simplified 97.3%

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

4. Final simplification 98.5%

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

Alternative 5: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ (/ 0.5 x) (+ x x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (x + x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(x + x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (x + x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.8%

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

   2. Taylor expanded in x around -inf 98.4%

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

      1. sub-neg 98.4%

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

      2. neg-mul-1 98.4%

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

      3. associate-+r+ 98.3%

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

      4. rem-square-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. rem-square-sqrt 10.3%

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

      7. +-commutative 10.3%

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

      8. unsub-neg 10.3%

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

      9. associate-*r/ 10.3%

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

      10. metadata-eval 10.3%

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

      11. distribute-neg-frac 10.3%

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

      12. metadata-eval 10.3%

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

   4. Simplified 10.3%

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

   5. Taylor expanded in x around 0 98.2%

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

   if -10 < x < 1

   1. Initial program 24.9%

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

      1. flip-+ 25.0%

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

      2. sqr-abs 25.0%

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

      3. add-sqr-sqrt 24.9%

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

      4. div-sub 25.0%

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

      5. add-sqr-sqrt 12.4%

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

      6. fabs-sqr 12.4%

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

      7. add-sqr-sqrt 23.0%

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

      8. +-commutative 23.0%

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

      9. hypot-1-def 23.0%

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

      10. fma-def 22.9%

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

      11. add-sqr-sqrt 12.4%

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

      12. fabs-sqr 12.4%

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

      13. add-sqr-sqrt 25.0%

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

      14. +-commutative 25.0%

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

      15. hypot-1-def 24.9%

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

   3. Applied egg-rr 24.9%

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

      1. div-sub 24.9%

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

      2. *-lft-identity 24.9%

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

      3. metadata-eval 24.9%

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

      4. times-frac 24.9%

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

      5. fma-udef 24.9%

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

      6. associate--r+ 25.0%

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

      7. +-inverses 25.0%

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

      8. metadata-eval 25.0%

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

      9. metadata-eval 25.0%

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

      10. neg-mul-1 25.0%

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

      11. neg-sub0 25.0%

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

      12. associate--r- 25.0%

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

      13. neg-sub0 25.0%

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

      14. +-commutative 25.0%

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

      15. sub-neg 25.0%

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

   5. Simplified 25.0%

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

   6. Taylor expanded in x around 0 98.3%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

      3. rem-square-sqrt 95.0%

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

      4. fabs-sqr 95.0%

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

      5. rem-square-sqrt 95.1%

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

   4. Simplified 95.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 6: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 - \frac{0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ 1.0 (- (* x -2.0) (/ 0.5 x)))) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ (/ 0.5 x) (+ x x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((1.0f / ((x * -2.0f) - (0.5f / x)))), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (x + x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(Float32(x * Float32(-2.0)) - Float32(Float32(0.5) / x)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(x + x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(1.0) / ((x * single(-2.0)) - (single(0.5) / x)))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (x + x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.8%

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

      1. flip-+ 5.3%

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

      2. sqr-abs 5.3%

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

      3. add-sqr-sqrt 4.7%

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

      4. div-sub 4.8%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 1.7%

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

      8. +-commutative 1.7%

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

      9. hypot-1-def 1.7%

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

      10. fma-def 1.7%

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

      11. add-sqr-sqrt -0.0%

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

      12. fabs-sqr -0.0%

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

      13. add-sqr-sqrt 8.1%

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

      14. +-commutative 8.1%

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

      15. hypot-1-def 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. div-sub 9.4%

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

      2. *-lft-identity 9.4%

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

      3. metadata-eval 9.4%

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

      4. times-frac 9.4%

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

      5. fma-udef 9.4%

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

      6. associate--r+ 53.9%

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

      7. +-inverses 98.5%

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

      8. metadata-eval 98.5%

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

      9. metadata-eval 98.5%

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

      10. neg-mul-1 98.5%

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

      11. neg-sub0 98.5%

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

      12. associate--r- 98.5%

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

      13. neg-sub0 98.5%

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

      14. +-commutative 98.5%

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

      15. sub-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around -inf 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*r/ 98.4%

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

      3. metadata-eval 98.4%

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

   8. Simplified 98.4%

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

   if -10 < x < 1

   1. Initial program 24.9%

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

      1. flip-+ 25.0%

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

      2. sqr-abs 25.0%

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

      3. add-sqr-sqrt 24.9%

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

      4. div-sub 25.0%

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

      5. add-sqr-sqrt 12.4%

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

      6. fabs-sqr 12.4%

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

      7. add-sqr-sqrt 23.0%

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

      8. +-commutative 23.0%

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

      9. hypot-1-def 23.0%

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

      10. fma-def 22.9%

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

      11. add-sqr-sqrt 12.4%

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

      12. fabs-sqr 12.4%

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

      13. add-sqr-sqrt 25.0%

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

      14. +-commutative 25.0%

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

      15. hypot-1-def 24.9%

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

   3. Applied egg-rr 24.9%

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

      1. div-sub 24.9%

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

      2. *-lft-identity 24.9%

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

      3. metadata-eval 24.9%

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

      4. times-frac 24.9%

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

      5. fma-udef 24.9%

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

      6. associate--r+ 25.0%

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

      7. +-inverses 25.0%

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

      8. metadata-eval 25.0%

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

      9. metadata-eval 25.0%

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

      10. neg-mul-1 25.0%

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

      11. neg-sub0 25.0%

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

      12. associate--r- 25.0%

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

      13. neg-sub0 25.0%

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

      14. +-commutative 25.0%

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

      15. sub-neg 25.0%

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

   5. Simplified 25.0%

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

   6. Taylor expanded in x around 0 98.3%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

      3. rem-square-sqrt 95.0%

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

      4. fabs-sqr 95.0%

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

      5. rem-square-sqrt 95.1%

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

   4. Simplified 95.1%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 - \frac{0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 7: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.8%

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

   2. Taylor expanded in x around -inf 98.4%

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

      1. sub-neg 98.4%

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

      2. neg-mul-1 98.4%

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

      3. associate-+r+ 98.3%

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

      4. rem-square-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. rem-square-sqrt 10.3%

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

      7. +-commutative 10.3%

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

      8. unsub-neg 10.3%

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

      9. associate-*r/ 10.3%

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

      10. metadata-eval 10.3%

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

      11. distribute-neg-frac 10.3%

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

      12. metadata-eval 10.3%

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

   4. Simplified 10.3%

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

   5. Taylor expanded in x around 0 98.2%

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

   if -10 < x < 1

   1. Initial program 24.9%

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

      1. flip-+ 25.0%

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

      2. sqr-abs 25.0%

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

      3. add-sqr-sqrt 24.9%

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

      4. div-sub 25.0%

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

      5. add-sqr-sqrt 12.4%

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

      6. fabs-sqr 12.4%

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

      7. add-sqr-sqrt 23.0%

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

      8. +-commutative 23.0%

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

      9. hypot-1-def 23.0%

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

      10. fma-def 22.9%

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

      11. add-sqr-sqrt 12.4%

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

      12. fabs-sqr 12.4%

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

      13. add-sqr-sqrt 25.0%

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

      14. +-commutative 25.0%

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

      15. hypot-1-def 24.9%

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

   3. Applied egg-rr 24.9%

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

      1. div-sub 24.9%

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

      2. *-lft-identity 24.9%

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

      3. metadata-eval 24.9%

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

      4. times-frac 24.9%

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

      5. fma-udef 24.9%

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

      6. associate--r+ 25.0%

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

      7. +-inverses 25.0%

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

      8. metadata-eval 25.0%

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

      9. metadata-eval 25.0%

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

      10. neg-mul-1 25.0%

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

      11. neg-sub0 25.0%

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

      12. associate--r- 25.0%

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

      13. neg-sub0 25.0%

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

      14. +-commutative 25.0%

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

      15. sub-neg 25.0%

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

   5. Simplified 25.0%

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

   6. Taylor expanded in x around 0 98.3%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 93.9%

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

      1. rem-square-sqrt 93.8%

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

      2. fabs-sqr 93.8%

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

      3. rem-square-sqrt 93.9%

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

   4. Simplified 93.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 8: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.8%

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

   2. Taylor expanded in x around -inf 98.4%

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

      1. sub-neg 98.4%

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

      2. neg-mul-1 98.4%

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

      3. associate-+r+ 98.3%

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

      4. rem-square-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. rem-square-sqrt 10.3%

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

      7. +-commutative 10.3%

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

      8. unsub-neg 10.3%

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

      9. associate-*r/ 10.3%

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

      10. metadata-eval 10.3%

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

      11. distribute-neg-frac 10.3%

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

      12. metadata-eval 10.3%

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

   4. Simplified 10.3%

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

   5. Taylor expanded in x around 0 98.2%

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

   if -10 < x < 1

   1. Initial program 24.9%

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

   2. Taylor expanded in x around 0 20.5%

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

      1. log1p-def 92.3%

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

      2. rem-square-sqrt 47.9%

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

      3. fabs-sqr 47.9%

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

      4. rem-square-sqrt 92.2%

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

   4. Simplified 92.2%

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

   5. Taylor expanded in x around 0 96.5%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 93.9%

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

      1. rem-square-sqrt 93.8%

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

      2. fabs-sqr 93.8%

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

      3. rem-square-sqrt 93.9%

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

   4. Simplified 93.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 9: 62.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + Float32(1.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 35.7%

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

   2. Taylor expanded in x around 0 28.5%

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

      1. log1p-def 76.1%

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

      2. rem-square-sqrt 31.7%

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

      3. fabs-sqr 31.7%

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

      4. rem-square-sqrt 61.2%

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

   4. Simplified 61.2%

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

   5. Taylor expanded in x around 0 67.8%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around 0 44.0%

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

      1. rem-square-sqrt 44.0%

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

      2. fabs-sqr 44.0%

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

      3. rem-square-sqrt 44.0%

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

   4. Simplified 44.0%

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

4. Final simplification 61.4%

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

Alternative 10: 75.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 35.7%

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

   2. Taylor expanded in x around 0 28.5%

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

      1. log1p-def 76.1%

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

      2. rem-square-sqrt 31.7%

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

      3. fabs-sqr 31.7%

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

      4. rem-square-sqrt 61.2%

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

   4. Simplified 61.2%

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

   5. Taylor expanded in x around 0 67.8%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 93.9%

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

      1. rem-square-sqrt 93.8%

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

      2. fabs-sqr 93.8%

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

      3. rem-square-sqrt 93.9%

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

   4. Simplified 93.9%

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

4. Final simplification 74.8%

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

Alternative 11: 62.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= 1.0f) {
		tmp = copysignf(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(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 35.7%

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

   2. Taylor expanded in x around 0 28.5%

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

      1. log1p-def 76.1%

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

      2. rem-square-sqrt 31.7%

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

      3. fabs-sqr 31.7%

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

      4. rem-square-sqrt 61.2%

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

   4. Simplified 61.2%

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

   5. Taylor expanded in x around 0 67.8%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around 0 44.0%

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

      1. log1p-def 44.0%

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

      2. rem-square-sqrt 44.0%

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

      3. fabs-sqr 44.0%

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

      4. rem-square-sqrt 44.0%

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

   4. Simplified 44.0%

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

4. Final simplification 61.4%

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

Alternative 12: 54.2% 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 40.6%

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

2. Taylor expanded in x around 0 32.6%

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

   1. log1p-def 67.4%

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

   2. rem-square-sqrt 35.0%

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

   3. fabs-sqr 35.0%

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

   4. rem-square-sqrt 56.5%

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

4. Simplified 56.5%

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

5. Taylor expanded in x around 0 52.6%

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

6. Final simplification 52.6%

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

Developer target: 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 2023174 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

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

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