Rust f32::asinh

Percentage Accurate: 38.4% → 99.5%

Time: 7.7s

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: 38.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05000000074505806))
		tmp = copysign(log(Float32(Float32(-1.0) / Float32(x - hypot(Float32(1.0), x)))), x);
	elseif (x <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x - Float32((x ^ Float32(3.0)) * Float32(0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.05000000074505806))
		tmp = sign(x) * abs(log((single(-1.0) / (x - hypot(single(1.0), x)))));
	elseif (x <= single(0.10000000149011612))
		tmp = sign(x) * abs((x - ((x ^ single(3.0)) * single(0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 55.7%

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

      1. flip-+ 9.2%

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

      2. clear-num 9.2%

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

      3. log-rec 9.2%

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

      4. +-commutative 9.2%

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

      5. hypot-1-def 9.2%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 8.5%

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

      9. sqr-abs 8.5%

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

      10. add-sqr-sqrt 11.2%

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

      11. +-commutative 11.2%

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

   3. Applied egg-rr 11.2%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. clear-num 99.8%

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

      2. log-rec 99.8%

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

   6. Applied egg-rr 99.8%

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

   if -0.0500000007 < x < 0.100000001

   1. Initial program 19.3%

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

      1. flip-+ 19.3%

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

      2. clear-num 19.2%

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

      3. log-rec 19.6%

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

      4. +-commutative 19.6%

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

      5. hypot-1-def 19.6%

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

      6. add-sqr-sqrt 11.7%

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

      7. fabs-sqr 11.7%

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

      8. add-sqr-sqrt 19.4%

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

      9. sqr-abs 19.4%

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

      10. add-sqr-sqrt 19.6%

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

      11. +-commutative 19.6%

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

   3. Applied egg-rr 19.6%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + \left(-1 \cdot x + -0.075 \cdot {x}^{5}\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + -1 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. *-commutative 99.9%

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

   7. Simplified 99.9%

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

   if 0.100000001 < x

   1. Initial program 54.6%

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

      1. *-un-lft-identity 54.6%

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

      2. *-commutative 54.6%

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

      3. log-prod 54.6%

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

      4. add-sqr-sqrt 54.6%

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

      5. fabs-sqr 54.6%

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

      6. add-sqr-sqrt 54.6%

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

      7. +-commutative 54.6%

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

      8. hypot-1-def 99.6%

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

      9. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= -1.0f) {
		tmp = copysignf(logf((-1.0f / (x - hypotf(1.0f, x)))), x);
	} else {
		tmp = copysignf(log1pf((x + (-1.0f + hypotf(1.0f, x)))), 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(-1.0))
		tmp = copysign(log(Float32(Float32(-1.0) / Float32(x - hypot(Float32(1.0), x)))), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(Float32(-1.0) + hypot(Float32(1.0), x)))), 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) < -1

   1. Initial program 55.2%

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

      1. flip-+ 8.0%

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

      2. clear-num 8.0%

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

      3. log-rec 8.0%

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

      4. +-commutative 8.0%

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

      5. hypot-1-def 8.0%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 7.3%

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

      9. sqr-abs 7.3%

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

      10. add-sqr-sqrt 10.1%

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

      11. +-commutative 10.1%

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

   3. Applied egg-rr 10.1%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. clear-num 99.9%

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

      2. log-rec 100.0%

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

   6. Applied egg-rr 100.0%

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

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

   1. Initial program 29.9%

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

      1. *-un-lft-identity 29.9%

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

      2. *-commutative 29.9%

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

      3. log-prod 29.9%

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

      4. add-sqr-sqrt 23.9%

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

      5. fabs-sqr 23.9%

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

      6. add-sqr-sqrt 30.0%

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

      7. +-commutative 30.0%

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

      8. hypot-1-def 42.9%

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

      9. metadata-eval 42.9%

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

   3. Applied egg-rr 42.9%

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

      1. log1p-expm1-u 42.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} + 0, x\right) \]

      2. expm1-udef 42.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} - 1}\right) + 0, x\right) \]

      3. add-exp-log 42.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(x + \mathsf{hypot}\left(1, x\right)\right)} - 1\right) + 0, x\right) \]

   5. Applied egg-rr 42.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)} + 0, x\right) \]
   6. Step-by-step derivation

      1. associate--l+ 98.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)}\right) + 0, x\right) \]

   7. Simplified 98.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)} + 0, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

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

Alternative 3: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05000000074505806))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x - Float32((x ^ Float32(3.0)) * Float32(0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.05000000074505806))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.10000000149011612))
		tmp = sign(x) * abs((x - ((x ^ single(3.0)) * single(0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 55.7%

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

      1. flip-+ 9.2%

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

      2. clear-num 9.2%

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

      3. log-rec 9.2%

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

      4. +-commutative 9.2%

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

      5. hypot-1-def 9.2%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 8.5%

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

      9. sqr-abs 8.5%

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

      10. add-sqr-sqrt 11.2%

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

      11. +-commutative 11.2%

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

   3. Applied egg-rr 11.2%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. log-prod 99.8%

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

      3. metadata-eval 99.8%

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

      4. frac-2neg 99.8%

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

      5. metadata-eval 99.8%

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

      6. /-rgt-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-lft-identity 99.8%

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

      2. neg-sub0 99.8%

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

      3. associate--r- 99.8%

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

      4. neg-sub0 99.8%

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

   8. Simplified 99.8%

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

   if -0.0500000007 < x < 0.100000001

   1. Initial program 19.3%

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

      1. flip-+ 19.3%

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

      2. clear-num 19.2%

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

      3. log-rec 19.6%

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

      4. +-commutative 19.6%

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

      5. hypot-1-def 19.6%

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

      6. add-sqr-sqrt 11.7%

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

      7. fabs-sqr 11.7%

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

      8. add-sqr-sqrt 19.4%

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

      9. sqr-abs 19.4%

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

      10. add-sqr-sqrt 19.6%

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

      11. +-commutative 19.6%

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

   3. Applied egg-rr 19.6%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + \left(-1 \cdot x + -0.075 \cdot {x}^{5}\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + -1 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. *-commutative 99.9%

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

   7. Simplified 99.9%

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

   if 0.100000001 < x

   1. Initial program 54.6%

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

      1. *-un-lft-identity 54.6%

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

      2. *-commutative 54.6%

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

      3. log-prod 54.6%

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

      4. add-sqr-sqrt 54.6%

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

      5. fabs-sqr 54.6%

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

      6. add-sqr-sqrt 54.6%

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

      7. +-commutative 54.6%

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

      8. hypot-1-def 99.6%

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

      9. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

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

Alternative 4: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05000000074505806)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.5)
     (copysign (- x (* (pow x 3.0) 0.16666666666666666)) x)
     (copysign (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.05000000074505806f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x - (powf(x, 3.0f) * 0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf(((x * 2.0f) + (0.5f * (1.0f / x)))), 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(0.5))
		tmp = copysign(Float32(x - Float32((x ^ Float32(3.0)) * Float32(0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(x * Float32(2.0)) + Float32(Float32(0.5) * Float32(Float32(1.0) / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.05000000074505806))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x - ((x ^ single(3.0)) * single(0.16666666666666666))));
	else
		tmp = sign(x) * abs(log(((x * single(2.0)) + (single(0.5) * (single(1.0) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 55.7%

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

      1. flip-+ 9.2%

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

      2. clear-num 9.2%

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

      3. log-rec 9.2%

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

      4. +-commutative 9.2%

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

      5. hypot-1-def 9.2%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 8.5%

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

      9. sqr-abs 8.5%

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

      10. add-sqr-sqrt 11.2%

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

      11. +-commutative 11.2%

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

   3. Applied egg-rr 11.2%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. log-prod 99.8%

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

      3. metadata-eval 99.8%

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

      4. frac-2neg 99.8%

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

      5. metadata-eval 99.8%

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

      6. /-rgt-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-lft-identity 99.8%

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

      2. neg-sub0 99.8%

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

      3. associate--r- 99.8%

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

      4. neg-sub0 99.8%

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

   8. Simplified 99.8%

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

   if -0.0500000007 < x < 0.5

   1. Initial program 21.6%

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

      1. flip-+ 21.5%

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

      2. clear-num 21.5%

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

      3. log-rec 21.9%

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

      4. +-commutative 21.9%

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

      5. hypot-1-def 21.8%

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

      6. add-sqr-sqrt 14.2%

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

      7. fabs-sqr 14.2%

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

      8. add-sqr-sqrt 21.6%

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

      9. sqr-abs 21.6%

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

      10. add-sqr-sqrt 21.8%

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

      11. +-commutative 21.8%

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

   3. Applied egg-rr 21.8%

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

   4. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + \left(-1 \cdot x + -0.075 \cdot {x}^{5}\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + -1 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   if 0.5 < x

   1. Initial program 51.3%

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

      1. *-un-lft-identity 51.3%

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

      2. *-commutative 51.3%

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

      3. log-prod 51.3%

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

      4. add-sqr-sqrt 51.3%

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

      5. fabs-sqr 51.3%

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

      6. add-sqr-sqrt 51.3%

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

      7. +-commutative 51.3%

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

      8. hypot-1-def 99.9%

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

      9. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 99.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \]

Alternative 5: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x - Float32((x ^ Float32(3.0)) * Float32(0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(x * Float32(2.0)) + Float32(Float32(0.5) * Float32(Float32(1.0) / 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(log((single(-0.5) / x)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x - ((x ^ single(3.0)) * single(0.16666666666666666))));
	else
		tmp = sign(x) * abs(log(((x * single(2.0)) + (single(0.5) * (single(1.0) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.2%

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

      1. *-un-lft-identity 55.2%

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

      2. *-commutative 55.2%

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

      3. log-prod 55.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 13.3%

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

      7. +-commutative 13.3%

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

      8. hypot-1-def 13.3%

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

      9. metadata-eval 13.3%

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

   3. Applied egg-rr 13.3%

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

   4. Taylor expanded in x around -inf 97.1%

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

   if -1 < x < 0.5

   1. Initial program 22.1%

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

      1. flip-+ 22.0%

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

      2. clear-num 22.0%

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

      3. log-rec 22.4%

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

      4. +-commutative 22.4%

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

      5. hypot-1-def 22.3%

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

      6. add-sqr-sqrt 14.1%

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

      7. fabs-sqr 14.1%

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

      8. add-sqr-sqrt 22.1%

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

      9. sqr-abs 22.1%

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

      10. add-sqr-sqrt 22.3%

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

      11. +-commutative 22.3%

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

   3. Applied egg-rr 22.3%

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

   4. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + \left(-1 \cdot x + -0.075 \cdot {x}^{5}\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + -1 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 98.9%

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

      2. unsub-neg 98.9%

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

      3. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   if 0.5 < x

   1. Initial program 51.3%

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

      1. *-un-lft-identity 51.3%

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

      2. *-commutative 51.3%

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

      3. log-prod 51.3%

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

      4. add-sqr-sqrt 51.3%

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

      5. fabs-sqr 51.3%

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

      6. add-sqr-sqrt 51.3%

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

      7. +-commutative 51.3%

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

      8. hypot-1-def 99.9%

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

      9. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 99.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

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

Alternative 6: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.2%

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

      1. *-un-lft-identity 55.2%

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

      2. *-commutative 55.2%

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

      3. log-prod 55.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 13.3%

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

      7. +-commutative 13.3%

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

      8. hypot-1-def 13.3%

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

      9. metadata-eval 13.3%

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

   3. Applied egg-rr 13.3%

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

   4. Taylor expanded in x around -inf 97.1%

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

   if -1 < x < 0.5

   1. Initial program 22.1%

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

      1. flip-+ 22.0%

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

      2. clear-num 22.0%

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

      3. log-rec 22.4%

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

      4. +-commutative 22.4%

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

      5. hypot-1-def 22.3%

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

      6. add-sqr-sqrt 14.1%

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

      7. fabs-sqr 14.1%

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

      8. add-sqr-sqrt 22.1%

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

      9. sqr-abs 22.1%

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

      10. add-sqr-sqrt 22.3%

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

      11. +-commutative 22.3%

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

   3. Applied egg-rr 22.3%

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

   4. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + \left(-1 \cdot x + -0.075 \cdot {x}^{5}\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + -1 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 98.9%

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

      2. unsub-neg 98.9%

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

      3. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   if 0.5 < x

   1. Initial program 51.3%

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

      1. *-un-lft-identity 51.3%

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

      2. *-commutative 51.3%

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

      3. log-prod 51.3%

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

      4. add-sqr-sqrt 51.3%

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

      5. fabs-sqr 51.3%

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

      6. add-sqr-sqrt 51.3%

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

      7. +-commutative 51.3%

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

      8. hypot-1-def 99.9%

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

      9. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 98.1%

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

      1. *-commutative 98.1%

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

   6. Simplified 98.1%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

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

Alternative 7: 70.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(Float32(-log(Float32(Float32(-1.0) / x))), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / 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)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(-log((single(1.0) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 54.6%

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

   2. Taylor expanded in x around -inf 44.1%

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

   if -10 < x < 0.5

   1. Initial program 22.6%

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

      1. flip-+ 22.6%

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

      2. clear-num 22.5%

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

      3. log-rec 22.9%

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

      4. +-commutative 22.9%

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

      5. hypot-1-def 22.9%

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

      6. add-sqr-sqrt 14.0%

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

      7. fabs-sqr 14.0%

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

      8. add-sqr-sqrt 22.7%

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

      9. sqr-abs 22.7%

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

      10. add-sqr-sqrt 22.9%

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

      11. +-commutative 22.9%

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

   3. Applied egg-rr 22.9%

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

   4. Taylor expanded in x around 0 97.0%

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

      1. neg-mul-1 97.0%

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

   6. Simplified 97.0%

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

   if 0.5 < x

   1. Initial program 51.3%

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

   2. Taylor expanded in x around inf 44.4%

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

4. Final simplification 72.7%

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

Alternative 8: 83.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 54.6%

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

   2. Taylor expanded in x around -inf 44.1%

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

   if -10 < x < 0.5

   1. Initial program 22.6%

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

      1. flip-+ 22.6%

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

      2. clear-num 22.5%

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

      3. log-rec 22.9%

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

      4. +-commutative 22.9%

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

      5. hypot-1-def 22.9%

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

      6. add-sqr-sqrt 14.0%

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

      7. fabs-sqr 14.0%

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

      8. add-sqr-sqrt 22.7%

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

      9. sqr-abs 22.7%

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

      10. add-sqr-sqrt 22.9%

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

      11. +-commutative 22.9%

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

   3. Applied egg-rr 22.9%

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

   4. Taylor expanded in x around 0 97.0%

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

      1. neg-mul-1 97.0%

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

   6. Simplified 97.0%

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

   if 0.5 < x

   1. Initial program 51.3%

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

      1. *-un-lft-identity 51.3%

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

      2. *-commutative 51.3%

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

      3. log-prod 51.3%

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

      4. add-sqr-sqrt 51.3%

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

      5. fabs-sqr 51.3%

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

      6. add-sqr-sqrt 51.3%

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

      7. +-commutative 51.3%

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

      8. hypot-1-def 99.9%

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

      9. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 98.1%

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

      1. *-commutative 98.1%

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

   6. Simplified 98.1%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.2%

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

Alternative 9: 97.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.2%

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

      1. *-un-lft-identity 55.2%

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

      2. *-commutative 55.2%

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

      3. log-prod 55.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 13.3%

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

      7. +-commutative 13.3%

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

      8. hypot-1-def 13.3%

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

      9. metadata-eval 13.3%

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

   3. Applied egg-rr 13.3%

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

   4. Taylor expanded in x around -inf 97.1%

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

   if -1 < x < 0.5

   1. Initial program 22.1%

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

      1. flip-+ 22.0%

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

      2. clear-num 22.0%

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

      3. log-rec 22.4%

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

      4. +-commutative 22.4%

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

      5. hypot-1-def 22.3%

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

      6. add-sqr-sqrt 14.1%

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

      7. fabs-sqr 14.1%

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

      8. add-sqr-sqrt 22.1%

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

      9. sqr-abs 22.1%

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

      10. add-sqr-sqrt 22.3%

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

      11. +-commutative 22.3%

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

   3. Applied egg-rr 22.3%

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

   4. Taylor expanded in x around 0 97.5%

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

      1. neg-mul-1 97.5%

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

   6. Simplified 97.5%

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

   if 0.5 < x

   1. Initial program 51.3%

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

      1. *-un-lft-identity 51.3%

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

      2. *-commutative 51.3%

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

      3. log-prod 51.3%

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

      4. add-sqr-sqrt 51.3%

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

      5. fabs-sqr 51.3%

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

      6. add-sqr-sqrt 51.3%

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

      7. +-commutative 51.3%

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

      8. hypot-1-def 99.9%

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

      9. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 98.1%

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

      1. *-commutative 98.1%

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

   6. Simplified 98.1%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

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

Alternative 10: 70.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(Float32(-log(Float32(Float32(-1.0) / x))), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(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)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 54.6%

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

   2. Taylor expanded in x around -inf 44.1%

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

   if -10 < x < 0.5

   1. Initial program 22.6%

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

      1. flip-+ 22.6%

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

      2. clear-num 22.5%

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

      3. log-rec 22.9%

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

      4. +-commutative 22.9%

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

      5. hypot-1-def 22.9%

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

      6. add-sqr-sqrt 14.0%

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

      7. fabs-sqr 14.0%

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

      8. add-sqr-sqrt 22.7%

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

      9. sqr-abs 22.7%

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

      10. add-sqr-sqrt 22.9%

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

      11. +-commutative 22.9%

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

   3. Applied egg-rr 22.9%

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

   4. Taylor expanded in x around 0 97.0%

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

      1. neg-mul-1 97.0%

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

   6. Simplified 97.0%

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

   if 0.5 < x

   1. Initial program 51.3%

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

   2. Taylor expanded in x around inf 44.4%

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

   3. Taylor expanded in x around 0 44.4%

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

      1. neg-mul-1 44.4%

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

   5. Simplified 44.4%

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

4. Final simplification 72.7%

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

Alternative 11: 62.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   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. flip-+ 17.3%

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

      2. clear-num 17.3%

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

      3. log-rec 17.6%

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

      4. +-commutative 17.6%

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

      5. hypot-1-def 17.5%

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

      6. add-sqr-sqrt 9.3%

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

      7. fabs-sqr 9.3%

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

      8. add-sqr-sqrt 17.2%

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

      9. sqr-abs 17.2%

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

      10. add-sqr-sqrt 18.2%

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

      11. +-commutative 18.2%

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

   3. Applied egg-rr 18.2%

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

   4. Taylor expanded in x around 0 68.7%

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

      1. neg-mul-1 68.7%

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

   6. Simplified 68.7%

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

   if 0.5 < x

   1. Initial program 51.3%

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

   2. Taylor expanded in x around inf 44.4%

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

   3. Taylor expanded in x around 0 44.4%

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

      1. neg-mul-1 44.4%

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

   5. Simplified 44.4%

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

4. Final simplification 64.0%

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

Alternative 12: 54.1% 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 36.7%

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

   1. flip-+ 14.9%

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

   2. clear-num 14.8%

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

   3. log-rec 15.0%

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

   4. +-commutative 15.0%

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

   5. hypot-1-def 15.0%

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

   6. add-sqr-sqrt 8.6%

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

   7. fabs-sqr 8.6%

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

   8. add-sqr-sqrt 14.7%

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

   9. sqr-abs 14.7%

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

   10. add-sqr-sqrt 15.6%

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

   11. +-commutative 15.6%

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

3. Applied egg-rr 15.6%

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

4. Taylor expanded in x around 0 57.4%

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

   1. neg-mul-1 57.4%

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

6. Simplified 57.4%

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

7. Final simplification 57.4%

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

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 2023189 
(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))