Rust f32::asinh

Percentage Accurate: 37.6% → 98.5%

Time: 8.5s

Alternatives: 16

Speedup: 3.8×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + 1\\ t_1 := {t_0}^{2}\\ t_2 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_2 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t_2 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{t_0} + \frac{3}{t_1}\right), \left(\frac{30}{{t_0}^{3}} + \left(\frac{45}{t_1} + \frac{45}{t_0}\right)\right) \cdot \left(0.001388888888888889 \cdot {x}^{6}\right)\right) + \mathsf{fma}\left(0.5, \frac{{x}^{2}}{t_0}, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ (fabs x) 1.0))
        (t_1 (pow t_0 2.0))
        (t_2 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_2 -5.0)
     (copysign (log (+ (- x x) (/ -0.5 x))) x)
     (if (<= t_2 0.10000000149011612)
       (copysign
        (+
         (fma
          -0.041666666666666664
          (* (pow x 4.0) (+ (/ 3.0 t_0) (/ 3.0 t_1)))
          (*
           (+ (/ 30.0 (pow t_0 3.0)) (+ (/ 45.0 t_1) (/ 45.0 t_0)))
           (* 0.001388888888888889 (pow x 6.0))))
         (fma 0.5 (/ (pow x 2.0) t_0) (log1p (fabs x))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = fabsf(x) + 1.0f;
	float t_1 = powf(t_0, 2.0f);
	float t_2 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_2 <= -5.0f) {
		tmp = copysignf(logf(((x - x) + (-0.5f / x))), x);
	} else if (t_2 <= 0.10000000149011612f) {
		tmp = copysignf((fmaf(-0.041666666666666664f, (powf(x, 4.0f) * ((3.0f / t_0) + (3.0f / t_1))), (((30.0f / powf(t_0, 3.0f)) + ((45.0f / t_1) + (45.0f / t_0))) * (0.001388888888888889f * powf(x, 6.0f)))) + fmaf(0.5f, (powf(x, 2.0f) / t_0), log1pf(fabsf(x)))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(abs(x) + Float32(1.0))
	t_1 = t_0 ^ Float32(2.0)
	t_2 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_2 <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(x - x) + Float32(Float32(-0.5) / x))), x);
	elseif (t_2 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(fma(Float32(-0.041666666666666664), Float32((x ^ Float32(4.0)) * Float32(Float32(Float32(3.0) / t_0) + Float32(Float32(3.0) / t_1))), Float32(Float32(Float32(Float32(30.0) / (t_0 ^ Float32(3.0))) + Float32(Float32(Float32(45.0) / t_1) + Float32(Float32(45.0) / t_0))) * Float32(Float32(0.001388888888888889) * (x ^ Float32(6.0))))) + fma(Float32(0.5), Float32((x ^ Float32(2.0)) / t_0), log1p(abs(x)))), 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) < -5

   1. Initial program 50.3%

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

   2. Taylor expanded in x around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. unsub-neg 98.5%

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

      4. rem-square-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. rem-square-sqrt 99.6%

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

      7. associate-*r/ 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-neg-frac 99.6%

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

      10. metadata-eval 99.6%

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

   4. Simplified 99.6%

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

   if -5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.100000001

   1. Initial program 23.4%

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

   2. Taylor expanded in x around 0 23.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + \left(-0.041666666666666664 \cdot \left({x}^{4} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \left(0.001388888888888889 \cdot \left({x}^{6} \cdot \left(30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 45 \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right)\right)}, x\right) \]
   3. Step-by-step derivation

      1. +-commutative 23.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-0.041666666666666664 \cdot \left({x}^{4} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \left(0.001388888888888889 \cdot \left({x}^{6} \cdot \left(30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 45 \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right)\right) + \log \left(1 + \left|x\right|\right)}, x\right) \]

      2. associate-+r+ 23.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\left(-0.041666666666666664 \cdot \left({x}^{4} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + 0.001388888888888889 \cdot \left({x}^{6} \cdot \left(30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 45 \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right)\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right)} + \log \left(1 + \left|x\right|\right), x\right) \]

   4. Simplified 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{1 + \left|x\right|} + \frac{3}{{\left(1 + \left|x\right|\right)}^{2}}\right), \left(\frac{30}{{\left(1 + \left|x\right|\right)}^{3}} + \left(\frac{45}{{\left(1 + \left|x\right|\right)}^{2}} + \frac{45}{1 + \left|x\right|}\right)\right) \cdot \left(0.001388888888888889 \cdot {x}^{6}\right)\right) + \mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

   if 0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 46.4%

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

      1. *-un-lft-identity 46.4%

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

      2. *-commutative 46.4%

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

      3. log-prod 46.4%

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

      4. +-commutative 46.4%

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

      5. hypot-1-def 99.8%

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

      6. add-sqr-sqrt 99.9%

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

      7. fabs-sqr 99.9%

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

      8. add-sqr-sqrt 99.8%

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

      9. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{\left|x\right| + 1} + \frac{3}{{\left(\left|x\right| + 1\right)}^{2}}\right), \left(\frac{30}{{\left(\left|x\right| + 1\right)}^{3}} + \left(\frac{45}{{\left(\left|x\right| + 1\right)}^{2}} + \frac{45}{\left|x\right| + 1}\right)\right) \cdot \left(0.001388888888888889 \cdot {x}^{6}\right)\right) + \mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{2}\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_1 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t_1 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{x + 1} + \frac{3}{t_0}\right), \left(0.001388888888888889 \cdot {x}^{6}\right) \cdot \left(\frac{30}{{\left(x + 1\right)}^{3}} + \left(\frac{45}{t_0} + \frac{45}{x + 1}\right)\right)\right) + \mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (pow (+ x 1.0) 2.0))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_1 -5.0)
     (copysign (log (+ (- x x) (/ -0.5 x))) x)
     (if (<= t_1 0.10000000149011612)
       (copysign
        (+
         (fma
          -0.041666666666666664
          (* (pow x 4.0) (+ (/ 3.0 (+ x 1.0)) (/ 3.0 t_0)))
          (*
           (* 0.001388888888888889 (pow x 6.0))
           (+
            (/ 30.0 (pow (+ x 1.0) 3.0))
            (+ (/ 45.0 t_0) (/ 45.0 (+ x 1.0))))))
         (fma 0.5 (/ (pow x 2.0) (+ x 1.0)) (log1p x)))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = powf((x + 1.0f), 2.0f);
	float t_1 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_1 <= -5.0f) {
		tmp = copysignf(logf(((x - x) + (-0.5f / x))), x);
	} else if (t_1 <= 0.10000000149011612f) {
		tmp = copysignf((fmaf(-0.041666666666666664f, (powf(x, 4.0f) * ((3.0f / (x + 1.0f)) + (3.0f / t_0))), ((0.001388888888888889f * powf(x, 6.0f)) * ((30.0f / powf((x + 1.0f), 3.0f)) + ((45.0f / t_0) + (45.0f / (x + 1.0f)))))) + fmaf(0.5f, (powf(x, 2.0f) / (x + 1.0f)), log1pf(x))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(x + Float32(1.0)) ^ Float32(2.0)
	t_1 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_1 <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(x - x) + Float32(Float32(-0.5) / x))), x);
	elseif (t_1 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(fma(Float32(-0.041666666666666664), Float32((x ^ Float32(4.0)) * Float32(Float32(Float32(3.0) / Float32(x + Float32(1.0))) + Float32(Float32(3.0) / t_0))), Float32(Float32(Float32(0.001388888888888889) * (x ^ Float32(6.0))) * Float32(Float32(Float32(30.0) / (Float32(x + Float32(1.0)) ^ Float32(3.0))) + Float32(Float32(Float32(45.0) / t_0) + Float32(Float32(45.0) / Float32(x + Float32(1.0))))))) + fma(Float32(0.5), Float32((x ^ Float32(2.0)) / Float32(x + Float32(1.0))), log1p(x))), 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) < -5

   1. Initial program 50.3%

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

   2. Taylor expanded in x around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. unsub-neg 98.5%

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

      4. rem-square-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. rem-square-sqrt 99.6%

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

      7. associate-*r/ 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-neg-frac 99.6%

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

      10. metadata-eval 99.6%

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

   4. Simplified 99.6%

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

   if -5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.100000001

   1. Initial program 23.4%

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

   2. Taylor expanded in x around 0 23.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + \left(-0.041666666666666664 \cdot \left({x}^{4} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \left(0.001388888888888889 \cdot \left({x}^{6} \cdot \left(30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 45 \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right)\right)}, x\right) \]

   4. Simplified 99.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{1 + x} + \frac{3}{{\left(1 + x\right)}^{2}}\right), \left(\frac{30}{{\left(1 + x\right)}^{3}} + \left(\frac{45}{{\left(1 + x\right)}^{2}} + \frac{45}{1 + x}\right)\right) \cdot \left(0.001388888888888889 \cdot {x}^{6}\right)\right) + \mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + x}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   if 0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 46.4%

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

      1. *-un-lft-identity 46.4%

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

      2. *-commutative 46.4%

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

      3. log-prod 46.4%

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

      4. +-commutative 46.4%

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

      5. hypot-1-def 99.8%

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

      6. add-sqr-sqrt 99.9%

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

      7. fabs-sqr 99.9%

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

      8. add-sqr-sqrt 99.8%

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

      9. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{x + 1} + \frac{3}{{\left(x + 1\right)}^{2}}\right), \left(0.001388888888888889 \cdot {x}^{6}\right) \cdot \left(\frac{30}{{\left(x + 1\right)}^{3}} + \left(\frac{45}{{\left(x + 1\right)}^{2}} + \frac{45}{x + 1}\right)\right)\right) + \mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ \mathbf{if}\;t_0 \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{t_1}, \mathsf{log1p}\left(\left|x\right|\right)\right) + \left(\frac{3}{t_1} + \frac{3}{{t_1}^{2}}\right) \cdot \left(-0.041666666666666664 \cdot {x}^{4}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

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 -0.10000000149011612)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 0.10000000149011612)
       (copysign
        (+
         (fma 0.5 (/ (pow x 2.0) t_1) (log1p (fabs x)))
         (*
          (+ (/ 3.0 t_1) (/ 3.0 (pow t_1 2.0)))
          (* -0.041666666666666664 (pow x 4.0))))
        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 t_1 = fabsf(x) + 1.0f;
	float tmp;
	if (t_0 <= -0.10000000149011612f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = copysignf((fmaf(0.5f, (powf(x, 2.0f) / t_1), log1pf(fabsf(x))) + (((3.0f / t_1) + (3.0f / powf(t_1, 2.0f))) * (-0.041666666666666664f * powf(x, 4.0f)))), 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)
	t_1 = Float32(abs(x) + Float32(1.0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.10000000149011612))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(fma(Float32(0.5), Float32((x ^ Float32(2.0)) / t_1), log1p(abs(x))) + Float32(Float32(Float32(Float32(3.0) / t_1) + Float32(Float32(3.0) / (t_1 ^ Float32(2.0)))) * Float32(Float32(-0.041666666666666664) * (x ^ Float32(4.0))))), 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) < -0.100000001

   1. Initial program 53.0%

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

      1. +-commutative 53.0%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

   if -0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.100000001

   1. Initial program 21.0%

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

   2. Taylor expanded in x around 0 21.8%

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

      1. +-commutative 21.8%

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

      2. associate-+r+ 21.8%

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

      3. +-commutative 21.8%

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

      4. fma-def 21.8%

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

      5. log1p-def 99.9%

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

      6. associate-*r* 99.9%

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

      7. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   if 0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 46.4%

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

      1. *-un-lft-identity 46.4%

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

      2. *-commutative 46.4%

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

      3. log-prod 46.4%

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

      4. +-commutative 46.4%

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

      5. hypot-1-def 99.8%

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

      6. add-sqr-sqrt 99.9%

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

      7. fabs-sqr 99.9%

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

      8. add-sqr-sqrt 99.8%

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

      9. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right) + \left(\frac{3}{\left|x\right| + 1} + \frac{3}{{\left(\left|x\right| + 1\right)}^{2}}\right) \cdot \left(-0.041666666666666664 \cdot {x}^{4}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right) + \left(\frac{3}{x + 1} + \frac{3}{{\left(x + 1\right)}^{2}}\right) \cdot \left(-0.041666666666666664 \cdot {x}^{4}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.10000000149011612)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 0.10000000149011612)
       (copysign
        (+
         (fma 0.5 (/ (pow x 2.0) (+ x 1.0)) (log1p x))
         (*
          (+ (/ 3.0 (+ x 1.0)) (/ 3.0 (pow (+ x 1.0) 2.0)))
          (* -0.041666666666666664 (pow x 4.0))))
        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 <= -0.10000000149011612f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = copysignf((fmaf(0.5f, (powf(x, 2.0f) / (x + 1.0f)), log1pf(x)) + (((3.0f / (x + 1.0f)) + (3.0f / powf((x + 1.0f), 2.0f))) * (-0.041666666666666664f * powf(x, 4.0f)))), 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(-0.10000000149011612))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(fma(Float32(0.5), Float32((x ^ Float32(2.0)) / Float32(x + Float32(1.0))), log1p(x)) + Float32(Float32(Float32(Float32(3.0) / Float32(x + Float32(1.0))) + Float32(Float32(3.0) / (Float32(x + Float32(1.0)) ^ Float32(2.0)))) * Float32(Float32(-0.041666666666666664) * (x ^ Float32(4.0))))), 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) < -0.100000001

   1. Initial program 53.0%

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

      1. +-commutative 53.0%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

   if -0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.100000001

   1. Initial program 21.0%

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

   2. Taylor expanded in x around 0 21.8%

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

      1. +-commutative 21.8%

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

      2. associate-+r+ 21.8%

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

   4. Simplified 99.9%

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

   if 0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 46.4%

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

      1. *-un-lft-identity 46.4%

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

      2. *-commutative 46.4%

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

      3. log-prod 46.4%

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

      4. +-commutative 46.4%

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

      5. hypot-1-def 99.8%

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

      6. add-sqr-sqrt 99.9%

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

      7. fabs-sqr 99.9%

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

      8. add-sqr-sqrt 99.8%

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

      9. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right) + \left(\frac{3}{x + 1} + \frac{3}{{\left(x + 1\right)}^{2}}\right) \cdot \left(-0.041666666666666664 \cdot {x}^{4}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.019999999552965164)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 0.009999999776482582)
       (copysign (fma 0.5 (/ (pow x 2.0) (+ x 1.0)) (log1p x)) 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 <= -0.019999999552965164f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else if (t_0 <= 0.009999999776482582f) {
		tmp = copysignf(fmaf(0.5f, (powf(x, 2.0f) / (x + 1.0f)), log1pf(x)), 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(-0.019999999552965164))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	elseif (t_0 <= Float32(0.009999999776482582))
		tmp = copysign(fma(Float32(0.5), Float32((x ^ Float32(2.0)) / Float32(x + Float32(1.0))), log1p(x)), 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) < -0.0199999996

   1. Initial program 53.6%

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

      1. +-commutative 53.6%

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

      2. hypot-1-def 98.3%

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

   3. Simplified 98.3%

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

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

   1. Initial program 19.8%

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

   2. Taylor expanded in x around 0 20.6%

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

      1. +-commutative 20.6%

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

      2. fma-def 20.6%

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

      3. rem-square-sqrt 10.9%

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

      4. fabs-sqr 10.9%

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

      5. rem-square-sqrt 20.5%

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

      6. log1p-def 99.5%

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

      7. rem-square-sqrt 46.8%

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

      8. fabs-sqr 46.8%

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

      9. rem-square-sqrt 99.9%

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

   4. Simplified 99.9%

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

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

   1. Initial program 46.9%

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

      1. *-un-lft-identity 46.9%

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

      2. *-commutative 46.9%

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

      3. log-prod 46.9%

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

      4. +-commutative 46.9%

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

      5. hypot-1-def 99.7%

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

      6. add-sqr-sqrt 99.7%

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

      7. fabs-sqr 99.7%

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

      8. add-sqr-sqrt 99.7%

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

      9. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. +-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 6: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(-0.5 \cdot {x}^{3} + \left(0.225 \cdot {x}^{5} + x \cdot 3\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (+ (- x x) (/ -0.5 x))) x)
     (if (<= t_0 0.10000000149011612)
       (copysign
        (*
         0.3333333333333333
         (+ (* -0.5 (pow x 3.0)) (+ (* 0.225 (pow x 5.0)) (* x 3.0))))
        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 <= -5.0f) {
		tmp = copysignf(logf(((x - x) + (-0.5f / x))), x);
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = copysignf((0.3333333333333333f * ((-0.5f * powf(x, 3.0f)) + ((0.225f * powf(x, 5.0f)) + (x * 3.0f)))), 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(-5.0))
		tmp = copysign(log(Float32(Float32(x - x) + Float32(Float32(-0.5) / x))), x);
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(Float32(Float32(-0.5) * (x ^ Float32(3.0))) + Float32(Float32(Float32(0.225) * (x ^ Float32(5.0))) + Float32(x * Float32(3.0))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-5.0))
		tmp = sign(x) * abs(log(((x - x) + (single(-0.5) / x))));
	elseif (t_0 <= single(0.10000000149011612))
		tmp = sign(x) * abs((single(0.3333333333333333) * ((single(-0.5) * (x ^ single(3.0))) + ((single(0.225) * (x ^ single(5.0))) + (x * single(3.0))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -5

   1. Initial program 50.3%

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

   2. Taylor expanded in x around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. unsub-neg 98.5%

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

      4. rem-square-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. rem-square-sqrt 99.6%

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

      7. associate-*r/ 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-neg-frac 99.6%

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

      10. metadata-eval 99.6%

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

   4. Simplified 99.6%

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

   if -5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.100000001

   1. Initial program 23.4%

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

      1. add-cbrt-cube 23.4%

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

      2. pow1/3 23.4%

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

      3. log-pow 23.4%

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

      4. pow3 23.4%

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

      5. log-pow 23.4%

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

      6. +-commutative 23.4%

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

      7. hypot-1-def 23.3%

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

      8. add-sqr-sqrt 10.6%

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

      9. fabs-sqr 10.6%

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

      10. add-sqr-sqrt 23.7%

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

   3. Applied egg-rr 23.7%

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

   4. Taylor expanded in x around 0 98.5%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(-0.5 \cdot {x}^{3} + \left(0.225 \cdot {x}^{5} + 3 \cdot x\right)\right)}, x\right) \]

   if 0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 46.4%

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

      1. *-un-lft-identity 46.4%

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

      2. *-commutative 46.4%

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

      3. log-prod 46.4%

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

      4. +-commutative 46.4%

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

      5. hypot-1-def 99.8%

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

      6. add-sqr-sqrt 99.9%

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

      7. fabs-sqr 99.9%

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

      8. add-sqr-sqrt 99.8%

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

      9. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(-0.5 \cdot {x}^{3} + \left(0.225 \cdot {x}^{5} + x \cdot 3\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 7: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + {x}^{3} \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (+ (- x x) (/ -0.5 x))) x)
   (if (<= x 0.10000000149011612)
     (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 <= -50.0f) {
		tmp = copysignf(logf(((x - x) + (-0.5f / x))), x);
	} else if (x <= 0.10000000149011612f) {
		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(-50.0))
		tmp = copysign(log(Float32(Float32(x - x) + Float32(Float32(-0.5) / x))), x);
	elseif (x <= Float32(0.10000000149011612))
		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(-50.0))
		tmp = sign(x) * abs(log(((x - x) + (single(-0.5) / x))));
	elseif (x <= single(0.10000000149011612))
		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 < -50

   1. Initial program 50.3%

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

   2. Taylor expanded in x around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. unsub-neg 98.5%

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

      4. rem-square-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. rem-square-sqrt 99.6%

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

      7. associate-*r/ 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-neg-frac 99.6%

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

      10. metadata-eval 99.6%

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

   4. Simplified 99.6%

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

   if -50 < x < 0.100000001

   1. Initial program 23.4%

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

      1. add-cbrt-cube 23.4%

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

      2. pow1/3 23.4%

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

      3. log-pow 23.4%

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

      4. pow3 23.4%

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

      5. log-pow 23.4%

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

      6. +-commutative 23.4%

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

      7. hypot-1-def 23.3%

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

      8. add-sqr-sqrt 10.6%

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

      9. fabs-sqr 10.6%

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

      10. add-sqr-sqrt 23.7%

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

   3. Applied egg-rr 23.7%

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

   4. Taylor expanded in x around 0 97.9%

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

   if 0.100000001 < x

   1. Initial program 46.4%

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

      1. *-un-lft-identity 46.4%

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

      2. *-commutative 46.4%

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

      3. log-prod 46.4%

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

      4. +-commutative 46.4%

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

      5. hypot-1-def 99.8%

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

      6. add-sqr-sqrt 99.9%

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

      7. fabs-sqr 99.9%

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

      8. add-sqr-sqrt 99.8%

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

      9. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. +-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + {x}^{3} \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 8: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + {x}^{3} \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (+ (- x x) (/ -0.5 x))) x)
   (if (<= x 0.4000000059604645)
     (copysign
      (* 0.3333333333333333 (* 3.0 (+ x (* (pow x 3.0) -0.16666666666666666))))
      x)
     (copysign (log (+ (/ 0.5 x) (+ x x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf(((x - x) + (-0.5f / x))), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf((0.3333333333333333f * (3.0f * (x + (powf(x, 3.0f) * -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (x + x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50.0))
		tmp = copysign(log(Float32(Float32(x - x) + Float32(Float32(-0.5) / x))), x);
	elseif (x <= Float32(0.4000000059604645))
		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(Float32(0.5) / x) + Float32(x + x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-50.0))
		tmp = sign(x) * abs(log(((x - x) + (single(-0.5) / x))));
	elseif (x <= single(0.4000000059604645))
		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) + (x + x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -50

   1. Initial program 50.3%

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

   2. Taylor expanded in x around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. unsub-neg 98.5%

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

      4. rem-square-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. rem-square-sqrt 99.6%

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

      7. associate-*r/ 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-neg-frac 99.6%

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

      10. metadata-eval 99.6%

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

   4. Simplified 99.6%

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

   if -50 < x < 0.400000006

   1. Initial program 24.0%

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

      1. add-cbrt-cube 24.0%

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

      2. pow1/3 24.0%

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

      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 23.9%

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

      5. log-pow 24.0%

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

      6. +-commutative 24.0%

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

      7. hypot-1-def 23.9%

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

      8. add-sqr-sqrt 11.3%

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

      9. fabs-sqr 11.3%

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

      10. add-sqr-sqrt 24.3%

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

   3. Applied egg-rr 24.3%

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

   4. Taylor expanded in x around 0 97.6%

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

   if 0.400000006 < x

   1. Initial program 45.7%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. associate-+r+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-*r/ 99.4%

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

      4. metadata-eval 99.4%

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

      5. rem-square-sqrt 99.4%

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

      6. fabs-sqr 99.4%

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

      7. rem-square-sqrt 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(x + {x}^{3} \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 9: 96.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (+ (- x x) (/ -0.5 x))) x)
   (if (<= x 0.4000000059604645)
     (copysign (* 0.3333333333333333 (* x 3.0)) x)
     (copysign (log (+ (/ 0.5 x) (+ x x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf(((x - x) + (-0.5f / x))), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf((0.3333333333333333f * (x * 3.0f)), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (x + x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50.0))
		tmp = copysign(log(Float32(Float32(x - x) + Float32(Float32(-0.5) / x))), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(x * Float32(3.0))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(x + x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-50.0))
		tmp = sign(x) * abs(log(((x - x) + (single(-0.5) / x))));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs((single(0.3333333333333333) * (x * single(3.0))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (x + x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -50

   1. Initial program 50.3%

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

   2. Taylor expanded in x around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. unsub-neg 98.5%

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

      4. rem-square-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. rem-square-sqrt 99.6%

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

      7. associate-*r/ 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-neg-frac 99.6%

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

      10. metadata-eval 99.6%

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

   4. Simplified 99.6%

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

   if -50 < x < 0.400000006

   1. Initial program 24.0%

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

      1. add-cbrt-cube 24.0%

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

      2. pow1/3 24.0%

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

      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 23.9%

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

      5. log-pow 24.0%

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

      6. +-commutative 24.0%

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

      7. hypot-1-def 23.9%

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

      8. add-sqr-sqrt 11.3%

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

      9. fabs-sqr 11.3%

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

      10. add-sqr-sqrt 24.3%

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

   3. Applied egg-rr 24.3%

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

   4. Taylor expanded in x around 0 95.9%

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

   if 0.400000006 < x

   1. Initial program 45.7%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. associate-+r+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-*r/ 99.4%

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

      4. metadata-eval 99.4%

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

      5. rem-square-sqrt 99.4%

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

      6. fabs-sqr 99.4%

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

      7. rem-square-sqrt 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 10: 96.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (+ (- x x) (/ -0.5 x))) x)
   (if (<= x 0.4000000059604645)
     (copysign (* 0.3333333333333333 (* x 3.0)) x)
     (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf(((x - x) + (-0.5f / x))), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf((0.3333333333333333f * (x * 3.0f)), x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50.0))
		tmp = copysign(log(Float32(Float32(x - x) + Float32(Float32(-0.5) / x))), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(x * Float32(3.0))), x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-50.0))
		tmp = sign(x) * abs(log(((x - x) + (single(-0.5) / x))));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs((single(0.3333333333333333) * (x * single(3.0))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -50

   1. Initial program 50.3%

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

   2. Taylor expanded in x around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. unsub-neg 98.5%

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

      4. rem-square-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. rem-square-sqrt 99.6%

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

      7. associate-*r/ 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-neg-frac 99.6%

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

      10. metadata-eval 99.6%

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

   4. Simplified 99.6%

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

   if -50 < x < 0.400000006

   1. Initial program 24.0%

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

      1. add-cbrt-cube 24.0%

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

      2. pow1/3 24.0%

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

      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 23.9%

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

      5. log-pow 24.0%

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

      6. +-commutative 24.0%

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

      7. hypot-1-def 23.9%

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

      8. add-sqr-sqrt 11.3%

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

      9. fabs-sqr 11.3%

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

      10. add-sqr-sqrt 24.3%

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

   3. Applied egg-rr 24.3%

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

   4. Taylor expanded in x around 0 95.9%

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

   if 0.400000006 < x

   1. Initial program 45.7%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. rem-square-sqrt 98.5%

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

      2. fabs-sqr 98.5%

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

      3. rem-square-sqrt 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 11: 83.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.5)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 0.4000000059604645)
     (copysign (* 0.3333333333333333 (* x 3.0)) x)
     (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.5f) {
		tmp = copysignf(-logf((-1.0f / x)), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf((0.3333333333333333f * (x * 3.0f)), x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.5))
		tmp = copysign(Float32(-log(Float32(Float32(-1.0) / x))), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(x * Float32(3.0))), x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.5))
		tmp = sign(x) * abs(-log((single(-1.0) / x)));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs((single(0.3333333333333333) * (x * single(3.0))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 51.1%

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

   2. Taylor expanded in x around -inf 45.0%

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

      1. mul-1-neg 45.0%

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

   4. Simplified 45.0%

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

   if -0.5 < x < 0.400000006

   1. Initial program 23.4%

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

      1. add-cbrt-cube 23.4%

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

      2. pow1/3 23.4%

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

      3. log-pow 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 23.4%

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

      6. +-commutative 23.4%

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

      7. hypot-1-def 23.3%

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

      8. add-sqr-sqrt 11.4%

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

      9. fabs-sqr 11.4%

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

      10. add-sqr-sqrt 23.7%

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

   3. Applied egg-rr 23.7%

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

   4. Taylor expanded in x around 0 96.3%

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

   if 0.400000006 < x

   1. Initial program 45.7%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. rem-square-sqrt 98.5%

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

      2. fabs-sqr 98.5%

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

      3. rem-square-sqrt 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 12: 84.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(-1 - \log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (- -1.0 (log (- x))) x)
   (if (<= x 0.4000000059604645)
     (copysign (* 0.3333333333333333 (* x 3.0)) x)
     (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf((-1.0f - logf(-x)), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf((0.3333333333333333f * (x * 3.0f)), x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50.0))
		tmp = copysign(Float32(Float32(-1.0) - log(Float32(-x))), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(x * Float32(3.0))), x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-50.0))
		tmp = sign(x) * abs((single(-1.0) - log(-x)));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs((single(0.3333333333333333) * (x * single(3.0))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -50

   1. Initial program 50.3%

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

   2. Taylor expanded in x around inf 6.1%

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

      1. associate-+r+ -0.0%

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

      2. +-commutative -0.0%

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

      3. associate-*r/ -0.0%

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

      4. metadata-eval -0.0%

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

      5. rem-square-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. rem-square-sqrt -0.0%

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

   4. Simplified -0.0%

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

   6. Applied egg-rr -0.0%

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

   8. Simplified 48.6%

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

   if -50 < x < 0.400000006

   1. Initial program 24.0%

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

      1. add-cbrt-cube 24.0%

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

      2. pow1/3 24.0%

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

      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 23.9%

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

      5. log-pow 24.0%

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

      6. +-commutative 24.0%

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

      7. hypot-1-def 23.9%

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

      8. add-sqr-sqrt 11.3%

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

      9. fabs-sqr 11.3%

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

      10. add-sqr-sqrt 24.3%

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

   3. Applied egg-rr 24.3%

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

   4. Taylor expanded in x around 0 95.9%

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

   if 0.400000006 < x

   1. Initial program 45.7%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. rem-square-sqrt 98.5%

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

      2. fabs-sqr 98.5%

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

      3. rem-square-sqrt 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(-1 - \log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 13: 74.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.4000000059604645)
   (copysign (* 0.3333333333333333 (* x 3.0)) x)
   (copysign (log (+ x x)) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.4000000059604645f) {
		tmp = copysignf((0.3333333333333333f * (x * 3.0f)), x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.4000000059604645))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(x * Float32(3.0))), x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(0.4000000059604645))
		tmp = sign(x) * abs((single(0.3333333333333333) * (x * single(3.0))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.400000006

   1. Initial program 33.0%

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

      1. add-cbrt-cube 28.5%

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

      2. pow1/3 28.4%

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

      3. log-pow 28.3%

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

      4. pow3 28.3%

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

      5. log-pow 32.8%

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

      6. +-commutative 32.8%

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

      7. hypot-1-def 49.0%

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

      8. add-sqr-sqrt 7.5%

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

      9. fabs-sqr 7.5%

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

      10. add-sqr-sqrt 19.0%

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

   3. Applied egg-rr 19.0%

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

   4. Taylor expanded in x around 0 66.7%

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

   if 0.400000006 < x

   1. Initial program 45.7%

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

   2. Taylor expanded in x around inf 98.5%

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

      1. rem-square-sqrt 98.5%

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

      2. fabs-sqr 98.5%

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

      3. rem-square-sqrt 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 14: 61.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.4000000059604645)
   (copysign (* 0.3333333333333333 (* x 3.0)) x)
   (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.4000000059604645f) {
		tmp = copysignf((0.3333333333333333f * (x * 3.0f)), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.4000000059604645))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(x * Float32(3.0))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.400000006

   1. Initial program 33.0%

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

      1. add-cbrt-cube 28.5%

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

      2. pow1/3 28.4%

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

      3. log-pow 28.3%

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

      4. pow3 28.3%

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

      5. log-pow 32.8%

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

      6. +-commutative 32.8%

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

      7. hypot-1-def 49.0%

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

      8. add-sqr-sqrt 7.5%

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

      9. fabs-sqr 7.5%

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

      10. add-sqr-sqrt 19.0%

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

   3. Applied egg-rr 19.0%

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

   4. Taylor expanded in x around 0 66.7%

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

   if 0.400000006 < x

   1. Initial program 45.7%

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

   2. Taylor expanded in x around 0 44.5%

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

      1. log1p-def 44.5%

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

      2. rem-square-sqrt 44.5%

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

      3. fabs-sqr 44.5%

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

      4. rem-square-sqrt 44.5%

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

   4. Simplified 44.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 15: 53.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(0.3333333333333333 \cdot \left(x \cdot 3\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (* 0.3333333333333333 (* x 3.0)) x))

C

float code(float x) {
	return copysignf((0.3333333333333333f * (x * 3.0f)), x);
}

Julia

function code(x)
	return copysign(Float32(Float32(0.3333333333333333) * Float32(x * Float32(3.0))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((single(0.3333333333333333) * (x * single(3.0))));
end

Derivation

1. Initial program 36.4%

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

   1. add-cbrt-cube 30.2%

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

   2. pow1/3 30.1%

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

   3. log-pow 30.1%

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

   4. pow3 30.0%

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

   5. log-pow 36.2%

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

   6. +-commutative 36.2%

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

   7. hypot-1-def 62.5%

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

   8. add-sqr-sqrt 32.1%

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

   9. fabs-sqr 32.1%

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

   10. add-sqr-sqrt 40.5%

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

3. Applied egg-rr 40.5%

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

4. Taylor expanded in x around 0 51.7%

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

5. Final simplification 51.7%

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

Alternative 16: 18.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(0.3333333333333333 \cdot \left(-x\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (* 0.3333333333333333 (- x)) x))

C

float code(float x) {
	return copysignf((0.3333333333333333f * -x), x);
}

Julia

function code(x)
	return copysign(Float32(Float32(0.3333333333333333) * Float32(-x)), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((single(0.3333333333333333) * -x));
end

Derivation

1. Initial program 36.4%

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

   1. add-cbrt-cube 30.2%

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

   2. pow1/3 30.1%

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

   3. log-pow 30.1%

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

   4. pow3 30.0%

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

   5. log-pow 36.2%

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

   6. +-commutative 36.2%

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

   7. hypot-1-def 62.5%

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

   8. add-sqr-sqrt 32.1%

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

   9. fabs-sqr 32.1%

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

   10. add-sqr-sqrt 40.5%

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

3. Applied egg-rr 40.5%

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

4. Taylor expanded in x around 0 51.7%

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

6. Simplified 18.3%

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

7. Final simplification 18.3%

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

Developer target: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2023292 
(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))