Rust f32::asinh

Percentage Accurate: 38.2% → 99.4%

Time: 11.5s

Alternatives: 16

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 38.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{t_1} + \frac{3}{{t_1}^{2}}\right), 0.5 \cdot \frac{{x}^{2}}{t_1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (+ (fabs x) 1.0)))
   (if (<= t_0 -0.5)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (+
         (log1p (fabs x))
         (fma
          -0.041666666666666664
          (* (pow x 4.0) (+ (/ 3.0 t_1) (/ 3.0 (pow t_1 2.0))))
          (* 0.5 (/ (pow x 2.0) t_1))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = fabsf(x) + 1.0f;
	float tmp;
	if (t_0 <= -0.5f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf((log1pf(fabsf(x)) + fmaf(-0.041666666666666664f, (powf(x, 4.0f) * ((3.0f / t_1) + (3.0f / powf(t_1, 2.0f)))), (0.5f * (powf(x, 2.0f) / t_1)))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = Float32(abs(x) + Float32(1.0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.5))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(log1p(abs(x)) + fma(Float32(-0.041666666666666664), Float32((x ^ Float32(4.0)) * Float32(Float32(Float32(3.0) / t_1) + Float32(Float32(3.0) / (t_1 ^ Float32(2.0))))), Float32(Float32(0.5) * Float32((x ^ Float32(2.0)) / t_1)))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -0.5

   1. Initial program 58.7%

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

      1. +-commutative 58.7%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.0500000007

   1. Initial program 18.4%

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

   2. Taylor expanded in x around 0 19.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + \left(-0.041666666666666664 \cdot \left({x}^{4} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right)}, x\right) \]
   3. Step-by-step derivation

      1. log1p-def 99.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + \left(-0.041666666666666664 \cdot \left({x}^{4} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right), x\right) \]

      2. fma-def 99.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right), 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right)}, x\right) \]

      3. associate-*r/ 99.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\color{blue}{\frac{3 \cdot 1}{1 + \left|x\right|}} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right), 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right), x\right) \]

      4. metadata-eval 99.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{\color{blue}{3}}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right), 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right), x\right) \]

      5. associate-*r/ 99.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{1 + \left|x\right|} + \color{blue}{\frac{3 \cdot 1}{{\left(1 + \left|x\right|\right)}^{2}}}\right), 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right), x\right) \]

      6. metadata-eval 99.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{1 + \left|x\right|} + \frac{\color{blue}{3}}{{\left(1 + \left|x\right|\right)}^{2}}\right), 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right), x\right) \]

   4. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{1 + \left|x\right|} + \frac{3}{{\left(1 + \left|x\right|\right)}^{2}}\right), 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right)}, x\right) \]

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

   1. Initial program 52.5%

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

      1. *-un-lft-identity 52.5%

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

      2. *-commutative 52.5%

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

      3. log-prod 52.5%

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

      4. +-commutative 52.5%

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

      5. hypot-1-def 98.6%

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

      6. add-sqr-sqrt 98.6%

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

      7. fabs-sqr 98.6%

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

      8. add-sqr-sqrt 98.6%

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

      9. metadata-eval 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-rgt-identity 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{\left|x\right| + 1} + \frac{3}{{\left(\left|x\right| + 1\right)}^{2}}\right), 0.5 \cdot \frac{{x}^{2}}{\left|x\right| + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{\left|x\right| + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.03999999910593033)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 0.019999999552965164)
       (copysign
        (+ (log1p (fabs x)) (* 0.5 (/ (pow x 2.0) (+ (fabs x) 1.0))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.03999999910593033f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else if (t_0 <= 0.019999999552965164f) {
		tmp = copysignf((log1pf(fabsf(x)) + (0.5f * (powf(x, 2.0f) / (fabsf(x) + 1.0f)))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03999999910593033))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	elseif (t_0 <= Float32(0.019999999552965164))
		tmp = copysign(Float32(log1p(abs(x)) + Float32(Float32(0.5) * Float32((x ^ Float32(2.0)) / Float32(abs(x) + Float32(1.0))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -0.0399999991

   1. Initial program 59.2%

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

      1. +-commutative 59.2%

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

      2. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

   if -0.0399999991 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.0199999996

   1. Initial program 17.1%

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

   2. Taylor expanded in x around 0 18.1%

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

      1. log1p-def 99.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}, x\right) \]

   4. Simplified 99.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]

   if 0.0199999996 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 52.9%

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

      1. *-un-lft-identity 52.9%

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

      2. *-commutative 52.9%

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

      3. log-prod 52.9%

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

      4. +-commutative 52.9%

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

      5. hypot-1-def 98.4%

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

      6. add-sqr-sqrt 98.4%

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

      7. fabs-sqr 98.4%

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

      8. add-sqr-sqrt 98.4%

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

      9. metadata-eval 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-rgt-identity 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{\left|x\right| + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(x \cdot \frac{x}{x + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.009999999776482582)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 0.019999999552965164)
       (copysign (+ (log1p (fabs x)) (* 0.5 (* x (/ x (+ x 1.0))))) x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.009999999776482582f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else if (t_0 <= 0.019999999552965164f) {
		tmp = copysignf((log1pf(fabsf(x)) + (0.5f * (x * (x / (x + 1.0f))))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.009999999776482582))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	elseif (t_0 <= Float32(0.019999999552965164))
		tmp = copysign(Float32(log1p(abs(x)) + Float32(Float32(0.5) * Float32(x * Float32(x / Float32(x + Float32(1.0)))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -0.00999999978

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

      2. hypot-1-def 99.5%

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

   3. Simplified 99.5%

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

   if -0.00999999978 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.0199999996

   1. Initial program 15.8%

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

   2. Taylor expanded in x around 0 16.7%

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

      1. log1p-def 99.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}, x\right) \]

   4. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   5. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{1 + \left|x\right|}, x\right) \]

      2. *-un-lft-identity 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \frac{x \cdot x}{\color{blue}{1 \cdot \left(1 + \left|x\right|\right)}}, x\right) \]

      3. times-frac 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{x}{1 + \left|x\right|}\right)}, x\right) \]

      4. add-sqr-sqrt 53.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right), x\right) \]

      5. fabs-sqr 53.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right), x\right) \]

      6. add-sqr-sqrt 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{1 + \color{blue}{x}}\right), x\right) \]

      7. +-commutative 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{\color{blue}{x + 1}}\right), x\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{x}{x + 1}\right)}, x\right) \]

   if 0.0199999996 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 52.9%

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

      1. *-un-lft-identity 52.9%

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

      2. *-commutative 52.9%

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

      3. log-prod 52.9%

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

      4. +-commutative 52.9%

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

      5. hypot-1-def 98.4%

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

      6. add-sqr-sqrt 98.4%

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

      7. fabs-sqr 98.4%

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

      8. add-sqr-sqrt 98.4%

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

      9. metadata-eval 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-rgt-identity 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(x \cdot \frac{x}{x + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(x \cdot \frac{x}{x + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.009999999776482582)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 0.019999999552965164)
     (copysign (+ (log1p (fabs x)) (* 0.5 (* x (/ x (+ x 1.0))))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.009999999776482582f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (x <= 0.019999999552965164f) {
		tmp = copysignf((log1pf(fabsf(x)) + (0.5f * (x * (x / (x + 1.0f))))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.009999999776482582))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.019999999552965164))
		tmp = copysign(Float32(log1p(abs(x)) + Float32(Float32(0.5) * Float32(x * Float32(x / Float32(x + Float32(1.0)))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00999999978

   1. Initial program 59.9%

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

      1. add-cbrt-cube 42.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 42.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 42.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 42.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 59.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 59.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 98.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 20.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 20.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. flip-+ 19.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. unpow2 19.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      3. div-sub 17.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      4. hypot-udef 17.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      5. metadata-eval 17.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{\color{blue}{1} + x \cdot x} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      6. unpow2 17.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + \color{blue}{{x}^{2}}} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      7. hypot-udef 17.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. metadata-eval 17.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \sqrt{\color{blue}{1} + x \cdot x}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      9. unpow2 17.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      10. add-sqr-sqrt 17.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1 + {x}^{2}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      11. +-commutative 17.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{{x}^{2} + 1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   5. Applied egg-rr 17.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{{x}^{2} + 1}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. div-sub 20.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2} - \left({x}^{2} + 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. associate--r+ 56.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      3. +-inverses 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      4. metadata-eval 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      5. metadata-eval 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      6. metadata-eval 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{\color{blue}{--1}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      7. metadata-eval 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\color{blue}{\left(0 - 1\right)}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. +-inverses 56.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\left(\color{blue}{\left({x}^{2} - {x}^{2}\right)} - 1\right)}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      9. associate--r+ 20.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\color{blue}{\left({x}^{2} - \left({x}^{2} + 1\right)\right)}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      10. associate-/r* 20.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{-\left({x}^{2} - \left({x}^{2} + 1\right)\right)}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}\right), x\right) \]

      11. neg-mul-1 20.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-\left({x}^{2} - \left({x}^{2} + 1\right)\right)}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right)\right), x\right) \]

      12. neg-mul-1 20.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{-1 \cdot \left({x}^{2} - \left({x}^{2} + 1\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      13. associate--r+ 56.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \color{blue}{\left(\left({x}^{2} - {x}^{2}\right) - 1\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      14. +-inverses 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \left(\color{blue}{0} - 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      15. metadata-eval 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \color{blue}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      16. metadata-eval 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      17. neg-sub0 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right)\right), x\right) \]

      18. associate--r- 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right)\right), x\right) \]

      19. neg-sub0 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      20. +-commutative 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right)\right), x\right) \]

      21. sub-neg 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right)\right), x\right) \]

   7. Simplified 98.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}\right), x\right) \]
   8. Step-by-step derivation

      1. associate-*r* 99.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]

      2. metadata-eval 99.3%

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

      3. *-un-lft-identity 99.3%

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

   9. Applied egg-rr 99.3%

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

   if -0.00999999978 < x < 0.0199999996

   1. Initial program 15.8%

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

   2. Taylor expanded in x around 0 16.7%

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

      1. log1p-def 99.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}, x\right) \]

   4. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   5. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{1 + \left|x\right|}, x\right) \]

      2. *-un-lft-identity 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \frac{x \cdot x}{\color{blue}{1 \cdot \left(1 + \left|x\right|\right)}}, x\right) \]

      3. times-frac 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{x}{1 + \left|x\right|}\right)}, x\right) \]

      4. add-sqr-sqrt 53.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right), x\right) \]

      5. fabs-sqr 53.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right), x\right) \]

      6. add-sqr-sqrt 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{1 + \color{blue}{x}}\right), x\right) \]

      7. +-commutative 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(\frac{x}{1} \cdot \frac{x}{\color{blue}{x + 1}}\right), x\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{x}{x + 1}\right)}, x\right) \]

   if 0.0199999996 < x

   1. Initial program 52.9%

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

      1. *-un-lft-identity 52.9%

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

      2. *-commutative 52.9%

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

      3. log-prod 52.9%

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

      4. +-commutative 52.9%

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

      5. hypot-1-def 98.4%

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

      6. add-sqr-sqrt 98.4%

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

      7. fabs-sqr 98.4%

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

      8. add-sqr-sqrt 98.4%

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

      9. metadata-eval 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-rgt-identity 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \left(x \cdot \frac{x}{x + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 5: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{x \cdot -2 - \frac{0.5}{x}}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + {x}^{3} \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign
    (* 0.3333333333333333 (* 3.0 (log (/ 1.0 (- (* x -2.0) (/ 0.5 x))))))
    x)
   (if (<= x 0.05000000074505806)
     (copysign
      (* 0.3333333333333333 (* 3.0 (+ 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 <= -1.0f) {
		tmp = copysignf((0.3333333333333333f * (3.0f * logf((1.0f / ((x * -2.0f) - (0.5f / x)))))), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((0.3333333333333333f * (3.0f * (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(-1.0))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(Float32(3.0) * log(Float32(Float32(1.0) / Float32(Float32(x * Float32(-2.0)) - Float32(Float32(0.5) / x)))))), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(Float32(3.0) * 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(-1.0))
		tmp = sign(x) * abs((single(0.3333333333333333) * (single(3.0) * log((single(1.0) / ((x * single(-2.0)) - (single(0.5) / x)))))));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((single(0.3333333333333333) * (single(3.0) * (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 < -1

   1. Initial program 58.1%

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

      1. add-cbrt-cube 40.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 39.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 39.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 39.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 57.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 57.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 15.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 15.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. flip-+ 14.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. unpow2 14.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      3. div-sub 12.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      4. hypot-udef 12.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      5. metadata-eval 12.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{\color{blue}{1} + x \cdot x} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      6. unpow2 12.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + \color{blue}{{x}^{2}}} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      7. hypot-udef 12.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. metadata-eval 12.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \sqrt{\color{blue}{1} + x \cdot x}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      9. unpow2 12.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      10. add-sqr-sqrt 13.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1 + {x}^{2}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      11. +-commutative 13.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{{x}^{2} + 1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   5. Applied egg-rr 13.1%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{{x}^{2} + 1}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. div-sub 16.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2} - \left({x}^{2} + 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. associate--r+ 54.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      3. +-inverses 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      4. metadata-eval 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      5. metadata-eval 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      6. metadata-eval 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{\color{blue}{--1}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      7. metadata-eval 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\color{blue}{\left(0 - 1\right)}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. +-inverses 54.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\left(\color{blue}{\left({x}^{2} - {x}^{2}\right)} - 1\right)}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      9. associate--r+ 16.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\color{blue}{\left({x}^{2} - \left({x}^{2} + 1\right)\right)}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      10. associate-/r* 16.0%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{-\left({x}^{2} - \left({x}^{2} + 1\right)\right)}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}\right), x\right) \]

      11. neg-mul-1 16.0%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-\left({x}^{2} - \left({x}^{2} + 1\right)\right)}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right)\right), x\right) \]

      12. neg-mul-1 16.0%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{-1 \cdot \left({x}^{2} - \left({x}^{2} + 1\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      13. associate--r+ 54.6%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \color{blue}{\left(\left({x}^{2} - {x}^{2}\right) - 1\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      14. +-inverses 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \left(\color{blue}{0} - 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      15. metadata-eval 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \color{blue}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      16. metadata-eval 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      17. neg-sub0 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right)\right), x\right) \]

      18. associate--r- 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right)\right), x\right) \]

      19. neg-sub0 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      20. +-commutative 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right)\right), x\right) \]

      21. sub-neg 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right)\right), x\right) \]

   7. Simplified 98.9%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}\right), x\right) \]

   8. Taylor expanded in x around -inf 97.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{-2 \cdot x - 0.5 \cdot \frac{1}{x}}}\right)\right), x\right) \]
   9. Step-by-step derivation

      1. *-commutative 97.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{x \cdot -2} - 0.5 \cdot \frac{1}{x}}\right)\right), x\right) \]

      2. associate-*r/ 97.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{x \cdot -2 - \color{blue}{\frac{0.5 \cdot 1}{x}}}\right)\right), x\right) \]

      3. metadata-eval 97.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{x \cdot -2 - \frac{\color{blue}{0.5}}{x}}\right)\right), x\right) \]

   10. Simplified 97.4%

       \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{x \cdot -2 - \frac{0.5}{x}}}\right)\right), x\right) \]

   if -1 < x < 0.0500000007

   1. Initial program 19.1%

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

      1. add-cbrt-cube 19.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 19.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 19.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 19.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 19.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 19.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 19.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 9.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 9.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 19.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 19.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 98.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(x + -0.16666666666666666 \cdot {x}^{3}\right)}\right), x\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + \color{blue}{{x}^{3} \cdot -0.16666666666666666}\right)\right), x\right) \]

   6. Simplified 98.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(x + {x}^{3} \cdot -0.16666666666666666\right)}\right), x\right) \]

   if 0.0500000007 < x

   1. Initial program 52.5%

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

      1. *-un-lft-identity 52.5%

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

      2. *-commutative 52.5%

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

      3. log-prod 52.5%

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

      4. +-commutative 52.5%

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

      5. hypot-1-def 98.6%

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

      6. add-sqr-sqrt 98.6%

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

      7. fabs-sqr 98.6%

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

      8. add-sqr-sqrt 98.6%

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

      9. metadata-eval 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-rgt-identity 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{x \cdot -2 - \frac{0.5}{x}}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + {x}^{3} \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 6: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.0005000000237487257:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.009999999776482582)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 0.0005000000237487257)
     (copysign x x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.009999999776482582f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (x <= 0.0005000000237487257f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.009999999776482582))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.0005000000237487257))
		tmp = copysign(x, 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(-0.009999999776482582))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (x <= single(0.0005000000237487257))
		tmp = sign(x) * abs(x);
	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 < -0.00999999978

   1. Initial program 59.9%

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

      1. add-cbrt-cube 42.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 42.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 42.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 42.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 59.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 59.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 98.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 20.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 20.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. flip-+ 19.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. unpow2 19.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      3. div-sub 17.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      4. hypot-udef 17.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      5. metadata-eval 17.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{\color{blue}{1} + x \cdot x} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      6. unpow2 17.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + \color{blue}{{x}^{2}}} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      7. hypot-udef 17.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. metadata-eval 17.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \sqrt{\color{blue}{1} + x \cdot x}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      9. unpow2 17.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      10. add-sqr-sqrt 17.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1 + {x}^{2}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      11. +-commutative 17.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{{x}^{2} + 1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   5. Applied egg-rr 17.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{{x}^{2} + 1}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. div-sub 20.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2} - \left({x}^{2} + 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. associate--r+ 56.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      3. +-inverses 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      4. metadata-eval 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      5. metadata-eval 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      6. metadata-eval 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{\color{blue}{--1}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      7. metadata-eval 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\color{blue}{\left(0 - 1\right)}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. +-inverses 56.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\left(\color{blue}{\left({x}^{2} - {x}^{2}\right)} - 1\right)}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      9. associate--r+ 20.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\color{blue}{\left({x}^{2} - \left({x}^{2} + 1\right)\right)}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      10. associate-/r* 20.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{-\left({x}^{2} - \left({x}^{2} + 1\right)\right)}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}\right), x\right) \]

      11. neg-mul-1 20.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-\left({x}^{2} - \left({x}^{2} + 1\right)\right)}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right)\right), x\right) \]

      12. neg-mul-1 20.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{-1 \cdot \left({x}^{2} - \left({x}^{2} + 1\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      13. associate--r+ 56.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \color{blue}{\left(\left({x}^{2} - {x}^{2}\right) - 1\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      14. +-inverses 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \left(\color{blue}{0} - 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      15. metadata-eval 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \color{blue}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      16. metadata-eval 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      17. neg-sub0 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right)\right), x\right) \]

      18. associate--r- 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right)\right), x\right) \]

      19. neg-sub0 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      20. +-commutative 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right)\right), x\right) \]

      21. sub-neg 98.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right)\right), x\right) \]

   7. Simplified 98.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}\right), x\right) \]
   8. Step-by-step derivation

      1. associate-*r* 99.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]

      2. metadata-eval 99.3%

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

      3. *-un-lft-identity 99.3%

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

   9. Applied egg-rr 99.3%

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

   if -0.00999999978 < x < 5.00000024e-4

   1. Initial program 14.4%

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

      1. add-cbrt-cube 14.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 14.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 14.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 14.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 14.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 14.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 14.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 14.7%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 14.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   6. Simplified 98.9%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   7. Taylor expanded in x around 0 100.0%

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

   if 5.00000024e-4 < x

   1. Initial program 53.7%

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

      1. *-un-lft-identity 53.7%

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

      2. *-commutative 53.7%

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

      3. log-prod 53.7%

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

      4. +-commutative 53.7%

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

      5. hypot-1-def 98.1%

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

      6. add-sqr-sqrt 98.1%

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

      7. fabs-sqr 98.1%

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

      8. add-sqr-sqrt 98.1%

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

      9. metadata-eval 98.1%

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

   3. Applied egg-rr 98.1%

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

      1. +-rgt-identity 98.1%

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

   5. Simplified 98.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.0005000000237487257:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 7: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{x \cdot -2 - \frac{0.5}{x}}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + {x}^{3} \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign
    (* 0.3333333333333333 (* 3.0 (log (/ 1.0 (- (* x -2.0) (/ 0.5 x))))))
    x)
   (if (<= x 2.0)
     (copysign
      (* 0.3333333333333333 (* 3.0 (+ x (* (pow x 3.0) -0.16666666666666666))))
      x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf((0.3333333333333333f * (3.0f * logf((1.0f / ((x * -2.0f) - (0.5f / x)))))), x);
	} else if (x <= 2.0f) {
		tmp = copysignf((0.3333333333333333f * (3.0f * (x + (powf(x, 3.0f) * -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(Float32(3.0) * log(Float32(Float32(1.0) / Float32(Float32(x * Float32(-2.0)) - Float32(Float32(0.5) / x)))))), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(Float32(3.0) * Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs((single(0.3333333333333333) * (single(3.0) * log((single(1.0) / ((x * single(-2.0)) - (single(0.5) / x)))))));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs((single(0.3333333333333333) * (single(3.0) * (x + ((x ^ single(3.0)) * single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 58.1%

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

      1. add-cbrt-cube 40.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 39.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 39.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 39.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 57.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 57.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 15.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 15.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. flip-+ 14.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. unpow2 14.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      3. div-sub 12.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      4. hypot-udef 12.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      5. metadata-eval 12.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{\color{blue}{1} + x \cdot x} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      6. unpow2 12.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + \color{blue}{{x}^{2}}} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      7. hypot-udef 12.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. metadata-eval 12.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \sqrt{\color{blue}{1} + x \cdot x}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      9. unpow2 12.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 + {x}^{2}} \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      10. add-sqr-sqrt 13.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1 + {x}^{2}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      11. +-commutative 13.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{{x}^{2} + 1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   5. Applied egg-rr 13.1%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{{x}^{2} + 1}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. div-sub 16.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{{x}^{2} - \left({x}^{2} + 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. associate--r+ 54.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      3. +-inverses 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      4. metadata-eval 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      5. metadata-eval 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      6. metadata-eval 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{\color{blue}{--1}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      7. metadata-eval 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\color{blue}{\left(0 - 1\right)}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. +-inverses 54.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\left(\color{blue}{\left({x}^{2} - {x}^{2}\right)} - 1\right)}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      9. associate--r+ 16.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\frac{-\color{blue}{\left({x}^{2} - \left({x}^{2} + 1\right)\right)}}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      10. associate-/r* 16.0%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{-\left({x}^{2} - \left({x}^{2} + 1\right)\right)}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}\right), x\right) \]

      11. neg-mul-1 16.0%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-\left({x}^{2} - \left({x}^{2} + 1\right)\right)}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right)\right), x\right) \]

      12. neg-mul-1 16.0%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{-1 \cdot \left({x}^{2} - \left({x}^{2} + 1\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      13. associate--r+ 54.6%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \color{blue}{\left(\left({x}^{2} - {x}^{2}\right) - 1\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      14. +-inverses 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \left(\color{blue}{0} - 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      15. metadata-eval 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-1 \cdot \color{blue}{-1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      16. metadata-eval 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      17. neg-sub0 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right)\right), x\right) \]

      18. associate--r- 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right)\right), x\right) \]

      19. neg-sub0 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      20. +-commutative 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right)\right), x\right) \]

      21. sub-neg 98.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right)\right), x\right) \]

   7. Simplified 98.9%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}\right), x\right) \]

   8. Taylor expanded in x around -inf 97.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{-2 \cdot x - 0.5 \cdot \frac{1}{x}}}\right)\right), x\right) \]
   9. Step-by-step derivation

      1. *-commutative 97.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{x \cdot -2} - 0.5 \cdot \frac{1}{x}}\right)\right), x\right) \]

      2. associate-*r/ 97.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{x \cdot -2 - \color{blue}{\frac{0.5 \cdot 1}{x}}}\right)\right), x\right) \]

      3. metadata-eval 97.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{x \cdot -2 - \frac{\color{blue}{0.5}}{x}}\right)\right), x\right) \]

   10. Simplified 97.4%

       \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{\color{blue}{x \cdot -2 - \frac{0.5}{x}}}\right)\right), x\right) \]

   if -1 < x < 2

   1. Initial program 23.2%

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

      1. add-cbrt-cube 23.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 23.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 23.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 23.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 23.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 23.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 23.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 14.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 14.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 23.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 23.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 95.9%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(x + -0.16666666666666666 \cdot {x}^{3}\right)}\right), x\right) \]
   5. Step-by-step derivation

      1. *-commutative 95.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + \color{blue}{{x}^{3} \cdot -0.16666666666666666}\right)\right), x\right) \]

   6. Simplified 95.9%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(x + {x}^{3} \cdot -0.16666666666666666\right)}\right), x\right) \]

   if 2 < x

   1. Initial program 48.7%

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

      1. add-cbrt-cube 37.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 36.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 37.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 37.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 97.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around inf 97.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. log-rec 98.5%

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

      3. log-prod 99.1%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{1}{x \cdot -2 - \frac{0.5}{x}}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + {x}^{3} \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 8: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + {x}^{3} \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (* 0.3333333333333333 (* 3.0 (log (/ -0.5 x)))) x)
   (if (<= x 2.0)
     (copysign
      (* 0.3333333333333333 (* 3.0 (+ x (* (pow x 3.0) -0.16666666666666666))))
      x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf((0.3333333333333333f * (3.0f * logf((-0.5f / x)))), x);
	} else if (x <= 2.0f) {
		tmp = copysignf((0.3333333333333333f * (3.0f * (x + (powf(x, 3.0f) * -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(Float32(3.0) * log(Float32(Float32(-0.5) / x)))), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(Float32(3.0) * Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs((single(0.3333333333333333) * (single(3.0) * log((single(-0.5) / x)))));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs((single(0.3333333333333333) * (single(3.0) * (x + ((x ^ single(3.0)) * single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.5%

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

      1. add-cbrt-cube 39.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 38.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 38.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 38.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 56.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 56.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 14.7%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 14.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around -inf 96.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{-0.5}{x}\right)}\right), x\right) \]

   if -2 < x < 2

   1. Initial program 23.9%

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

      1. add-cbrt-cube 23.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 23.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 23.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 24.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 14.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 14.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 24.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 24.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 95.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(x + -0.16666666666666666 \cdot {x}^{3}\right)}\right), x\right) \]
   5. Step-by-step derivation

      1. *-commutative 95.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + \color{blue}{{x}^{3} \cdot -0.16666666666666666}\right)\right), x\right) \]

   6. Simplified 95.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(x + {x}^{3} \cdot -0.16666666666666666\right)}\right), x\right) \]

   if 2 < x

   1. Initial program 48.7%

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

      1. add-cbrt-cube 37.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 36.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 37.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 37.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 97.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around inf 97.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. log-rec 98.5%

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

      3. log-prod 99.1%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + {x}^{3} \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 9: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (* 0.3333333333333333 (* 3.0 (log (/ -0.5 x)))) x)
   (if (<= x 2.0) (copysign x x) (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf((0.3333333333333333f * (3.0f * logf((-0.5f / x)))), x);
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(Float32(3.0) * log(Float32(Float32(-0.5) / x)))), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs((single(0.3333333333333333) * (single(3.0) * log((single(-0.5) / x)))));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.5%

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

      1. add-cbrt-cube 39.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 38.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 38.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 38.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 56.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 56.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 98.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 14.7%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 14.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around -inf 96.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{-0.5}{x}\right)}\right), x\right) \]

   if -2 < x < 2

   1. Initial program 23.9%

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

      1. add-cbrt-cube 23.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 23.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 23.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 24.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 14.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 14.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 24.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 24.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 93.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 93.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   6. Simplified 93.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   7. Taylor expanded in x around 0 94.4%

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

   if 2 < x

   1. Initial program 48.7%

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

      1. add-cbrt-cube 37.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 36.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 37.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 37.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 97.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around inf 97.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. log-rec 98.5%

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

      3. log-prod 99.1%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 10: 84.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.5%

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

   2. Taylor expanded in x around -inf 43.7%

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

      1. mul-1-neg 43.7%

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

   4. Simplified 43.7%

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

   if -2 < x < 2

   1. Initial program 23.9%

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

      1. add-cbrt-cube 23.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 23.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 23.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 24.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 14.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 14.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 24.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 24.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 93.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 93.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   6. Simplified 93.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   7. Taylor expanded in x around 0 94.4%

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

   if 2 < x

   1. Initial program 48.7%

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

      1. add-cbrt-cube 37.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 36.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 37.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 37.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 97.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around inf 97.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. log-rec 98.5%

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

      3. log-prod 99.1%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 82.7%

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

Alternative 11: 84.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.5%

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

   2. Taylor expanded in x around -inf 43.7%

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

      1. mul-1-neg 43.7%

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

   4. Simplified 43.7%

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

      1. frac-2neg 43.7%

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

      2. metadata-eval 43.7%

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

      3. log-div 43.7%

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

      4. metadata-eval 43.7%

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

   6. Applied egg-rr 43.7%

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

      1. neg-sub0 43.7%

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

   8. Simplified 43.7%

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

   if -2 < x < 2

   1. Initial program 23.9%

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

      1. add-cbrt-cube 23.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 23.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 23.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 24.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 23.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 14.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 14.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 24.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 24.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 93.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 93.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   6. Simplified 93.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   7. Taylor expanded in x around 0 94.4%

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

   if 2 < x

   1. Initial program 48.7%

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

      1. add-cbrt-cube 37.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 36.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 37.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 37.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 97.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around inf 97.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. log-rec 98.5%

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

      3. log-prod 99.1%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 82.7%

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

Alternative 12: 75.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 36.0%

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

      1. add-cbrt-cube 29.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 29.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 29.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 29.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 35.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 35.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 50.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 8.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 8.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 20.7%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 20.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 64.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 64.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   6. Simplified 64.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   7. Taylor expanded in x around 0 64.6%

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

   if 2 < x

   1. Initial program 48.7%

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

      1. add-cbrt-cube 37.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 36.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 37.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 37.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 97.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around inf 97.1%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(2 \cdot x\right)}\right), x\right) \]
   5. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(x \cdot 2\right)}\right), x\right) \]

   6. Simplified 97.1%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(x \cdot 2\right)}\right), x\right) \]

   7. Taylor expanded in x around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. log-prod 97.9%

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

   9. Simplified 97.9%

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

4. Final simplification 74.1%

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

Alternative 13: 75.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 36.0%

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

      1. add-cbrt-cube 29.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 29.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 29.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 29.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 35.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 35.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 50.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 8.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 8.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 20.7%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 20.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 64.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 64.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   6. Simplified 64.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   7. Taylor expanded in x around 0 64.6%

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

   if 2 < x

   1. Initial program 48.7%

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

      1. add-cbrt-cube 37.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 36.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 37.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 37.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 48.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 97.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 97.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around inf 97.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. log-rec 98.5%

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

      3. log-prod 99.1%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 74.5%

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

Alternative 14: 62.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(2.0))
		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(2.0))
		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 < 2

   1. Initial program 36.0%

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

      1. add-cbrt-cube 29.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 29.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 29.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 29.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 35.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 35.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 50.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 8.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 8.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 20.7%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 20.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 64.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 64.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   6. Simplified 64.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   7. Taylor expanded in x around 0 64.6%

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

   if 2 < x

   1. Initial program 48.7%

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

   2. Taylor expanded in x around inf 44.6%

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

      1. mul-1-neg 44.6%

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

      2. log-rec 44.6%

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

      3. remove-double-neg 44.6%

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

   4. Simplified 44.6%

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

4. Final simplification 58.9%

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

Alternative 15: 62.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 2.0) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 36.0%

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

      1. add-cbrt-cube 29.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 29.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 29.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 29.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 35.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 35.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 50.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 8.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 8.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 20.7%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   3. Applied egg-rr 20.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 64.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 64.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   6. Simplified 64.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   7. Taylor expanded in x around 0 64.6%

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

   if 2 < x

   1. Initial program 48.7%

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

   2. Taylor expanded in x around 0 10.0%

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

      1. log1p-def 10.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}, x\right) \]

   4. Simplified 10.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]

   5. Taylor expanded in x around 0 44.6%

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

      1. log1p-def 44.6%

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

      2. rem-square-sqrt 44.6%

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

      3. fabs-sqr 44.6%

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

      4. rem-square-sqrt 44.6%

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

   7. Simplified 44.6%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 16: 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 39.6%

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

   1. add-cbrt-cube 31.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

   2. pow1/3 31.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

   3. log-pow 31.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   4. pow3 31.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

   5. log-pow 39.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   6. +-commutative 39.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

   7. hypot-1-def 64.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   8. add-sqr-sqrt 34.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   9. fabs-sqr 34.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   10. add-sqr-sqrt 42.7%

       \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

3. Applied egg-rr 42.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

4. Taylor expanded in x around 0 48.8%

   \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
5. Step-by-step derivation

   1. *-commutative 48.8%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

6. Simplified 48.8%

   \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

7. Taylor expanded in x around 0 49.3%

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

8. Final simplification 49.3%

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

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 2024024 
(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))