Rust f32::asinh

Percentage Accurate: 37.1% → 99.5%

Time: 6.5s

Alternatives: 15

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := {\left(x + 1\right)}^{2}\\ \mathbf{if}\;t_0 \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{x + 1} + \frac{3}{t_1}\right), {x}^{6} \cdot \left(\left(\frac{30}{{\left(x + 1\right)}^{3}} + \left(\frac{45}{t_1} + \frac{45}{x + 1}\right)\right) \cdot 0.001388888888888889\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (pow (+ x 1.0) 2.0)))
   (if (<= t_0 -0.4000000059604645)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (+
         (fma 0.5 (/ (pow x 2.0) (+ x 1.0)) (log1p x))
         (fma
          -0.041666666666666664
          (* (pow x 4.0) (+ (/ 3.0 (+ x 1.0)) (/ 3.0 t_1)))
          (*
           (pow x 6.0)
           (*
            (+
             (/ 30.0 (pow (+ x 1.0) 3.0))
             (+ (/ 45.0 t_1) (/ 45.0 (+ x 1.0))))
            0.001388888888888889))))
        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 = powf((x + 1.0f), 2.0f);
	float tmp;
	if (t_0 <= -0.4000000059604645f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf((fmaf(0.5f, (powf(x, 2.0f) / (x + 1.0f)), log1pf(x)) + fmaf(-0.041666666666666664f, (powf(x, 4.0f) * ((3.0f / (x + 1.0f)) + (3.0f / t_1))), (powf(x, 6.0f) * (((30.0f / powf((x + 1.0f), 3.0f)) + ((45.0f / t_1) + (45.0f / (x + 1.0f)))) * 0.001388888888888889f)))), 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(x + Float32(1.0)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.4000000059604645))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(fma(Float32(0.5), Float32((x ^ Float32(2.0)) / Float32(x + Float32(1.0))), log1p(x)) + fma(Float32(-0.041666666666666664), Float32((x ^ Float32(4.0)) * Float32(Float32(Float32(3.0) / Float32(x + Float32(1.0))) + Float32(Float32(3.0) / t_1))), Float32((x ^ Float32(6.0)) * Float32(Float32(Float32(Float32(30.0) / (Float32(x + Float32(1.0)) ^ Float32(3.0))) + Float32(Float32(Float32(45.0) / t_1) + Float32(Float32(45.0) / Float32(x + Float32(1.0))))) * Float32(0.001388888888888889))))), 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) < -0.400000006

   1. Initial program 49.1%

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

      1. +-commutative 49.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 9.9%

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

      2. frac-2neg 9.9%

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

      3. log-div 9.9%

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

   5. Applied egg-rr 14.4%

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

      1. sub-neg 14.4%

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

      2. sub-neg 14.4%

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

      3. fma-udef 14.4%

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

      4. unpow2 14.4%

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

      5. associate--r+ 45.6%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. neg-sub0 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-in 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -0.400000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.200000003

   1. Initial program 22.7%

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

      1. +-commutative 22.7%

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

      2. hypot-1-def 22.8%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in x around 0 23.4%

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

   6. Simplified 99.9%

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

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 64.9%

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

      1. +-commutative 64.9%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-un-lft-identity 100.0%

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

      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 100.0%

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

      9. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. +-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{x + 1} + \frac{3}{{\left(x + 1\right)}^{2}}\right), {x}^{6} \cdot \left(\left(\frac{30}{{\left(x + 1\right)}^{3}} + \left(\frac{45}{{\left(x + 1\right)}^{2}} + \frac{45}{x + 1}\right)\right) \cdot 0.001388888888888889\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ \mathbf{if}\;t_0 \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(0.5, \frac{{x}^{2}}{t_1}, -0.041666666666666664 \cdot \left({x}^{4} \cdot \left(\frac{3}{t_1} + \frac{3}{{t_1}^{2}}\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (+ (fabs x) 1.0)))
   (if (<= t_0 -0.4000000059604645)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (+
         (log1p (fabs x))
         (fma
          0.5
          (/ (pow x 2.0) t_1)
          (*
           -0.041666666666666664
           (* (pow x 4.0) (+ (/ 3.0 t_1) (/ 3.0 (pow t_1 2.0)))))))
        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 <= -0.4000000059604645f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf((log1pf(fabsf(x)) + fmaf(0.5f, (powf(x, 2.0f) / t_1), (-0.041666666666666664f * (powf(x, 4.0f) * ((3.0f / t_1) + (3.0f / powf(t_1, 2.0f))))))), 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(-0.4000000059604645))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(log1p(abs(x)) + fma(Float32(0.5), Float32((x ^ Float32(2.0)) / t_1), Float32(Float32(-0.041666666666666664) * Float32((x ^ Float32(4.0)) * Float32(Float32(Float32(3.0) / t_1) + Float32(Float32(3.0) / (t_1 ^ Float32(2.0)))))))), 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) < -0.400000006

   1. Initial program 49.1%

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

      1. +-commutative 49.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 9.9%

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

      2. frac-2neg 9.9%

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

      3. log-div 9.9%

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

   5. Applied egg-rr 14.4%

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

      1. sub-neg 14.4%

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

      2. sub-neg 14.4%

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

      3. fma-udef 14.4%

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

      4. unpow2 14.4%

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

      5. associate--r+ 45.6%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. neg-sub0 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-in 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -0.400000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.0500000007

   1. Initial program 22.2%

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

      1. +-commutative 22.2%

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

      2. hypot-1-def 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in x around 0 22.9%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*r/ 100.0%

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

      7. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

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

   1. Initial program 65.3%

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

      1. +-commutative 65.3%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. *-un-lft-identity 99.9%

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

      6. add-sqr-sqrt 99.6%

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

      7. fabs-sqr 99.6%

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

      8. add-sqr-sqrt 99.9%

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

      9. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. +-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|x\right| + 1}, -0.041666666666666664 \cdot \left({x}^{4} \cdot \left(\frac{3}{\left|x\right| + 1} + \frac{3}{{\left(\left|x\right| + 1\right)}^{2}}\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right) + {x}^{4} \cdot \left(-0.041666666666666664 \cdot \left(\frac{3}{x + 1} + \frac{3}{{\left(x + 1\right)}^{2}}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.4000000059604645)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (+
         (fma 0.5 (/ (pow x 2.0) (+ x 1.0)) (log1p x))
         (*
          (pow x 4.0)
          (*
           -0.041666666666666664
           (+ (/ 3.0 (+ x 1.0)) (/ 3.0 (pow (+ x 1.0) 2.0))))))
        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 <= -0.4000000059604645f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf((fmaf(0.5f, (powf(x, 2.0f) / (x + 1.0f)), log1pf(x)) + (powf(x, 4.0f) * (-0.041666666666666664f * ((3.0f / (x + 1.0f)) + (3.0f / powf((x + 1.0f), 2.0f)))))), 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(-0.4000000059604645))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(fma(Float32(0.5), Float32((x ^ Float32(2.0)) / Float32(x + Float32(1.0))), log1p(x)) + Float32((x ^ Float32(4.0)) * Float32(Float32(-0.041666666666666664) * Float32(Float32(Float32(3.0) / Float32(x + Float32(1.0))) + Float32(Float32(3.0) / (Float32(x + Float32(1.0)) ^ Float32(2.0))))))), 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) < -0.400000006

   1. Initial program 49.1%

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

      1. +-commutative 49.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 9.9%

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

      2. frac-2neg 9.9%

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

      3. log-div 9.9%

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

   5. Applied egg-rr 14.4%

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

      1. sub-neg 14.4%

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

      2. sub-neg 14.4%

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

      3. fma-udef 14.4%

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

      4. unpow2 14.4%

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

      5. associate--r+ 45.6%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. neg-sub0 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-in 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -0.400000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.0500000007

   1. Initial program 22.2%

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

      1. +-commutative 22.2%

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

      2. hypot-1-def 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in x around 0 22.9%

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

      1. +-commutative 22.9%

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

      2. associate-+r+ 22.9%

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

   6. Simplified 99.9%

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

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

   1. Initial program 65.3%

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

      1. +-commutative 65.3%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. *-un-lft-identity 99.9%

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

      6. add-sqr-sqrt 99.6%

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

      7. fabs-sqr 99.6%

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

      8. add-sqr-sqrt 99.9%

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

      9. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. +-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right) + {x}^{4} \cdot \left(-0.041666666666666664 \cdot \left(\frac{3}{x + 1} + \frac{3}{{\left(x + 1\right)}^{2}}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.03999999910593033)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.019999999552965164)
       (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 <= -0.03999999910593033f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.019999999552965164f) {
		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(-0.03999999910593033))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.019999999552965164))
		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) < -0.0399999991

   1. Initial program 49.8%

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

      1. +-commutative 49.8%

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

      2. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 11.3%

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

      2. frac-2neg 11.3%

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

      3. log-div 11.3%

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

   5. Applied egg-rr 15.7%

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

      1. sub-neg 15.7%

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

      2. sub-neg 15.7%

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

      3. fma-udef 15.7%

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

      4. unpow2 15.7%

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

      5. associate--r+ 46.4%

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

      6. +-inverses 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

      10. neg-sub0 99.8%

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

      11. sub-neg 99.8%

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

      12. +-commutative 99.8%

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

      13. distribute-neg-in 99.8%

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

      14. remove-double-neg 99.8%

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

      15. sub-neg 99.8%

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

   7. Simplified 99.8%

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

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

   1. Initial program 21.2%

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

      1. +-commutative 21.2%

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

      2. hypot-1-def 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in x around 0 21.9%

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

      1. +-commutative 21.9%

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

      2. fma-def 21.9%

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

      3. rem-square-sqrt 11.2%

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

      4. fabs-sqr 11.2%

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

      5. rem-square-sqrt 22.0%

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

      6. log1p-def 99.6%

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

      7. rem-square-sqrt 43.9%

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

      8. fabs-sqr 43.9%

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

      9. rem-square-sqrt 99.8%

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

   6. Simplified 99.8%

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

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

   1. Initial program 65.6%

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

      1. +-commutative 65.6%

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

      2. hypot-1-def 99.6%

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

   3. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. *-commutative 99.6%

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

      3. log-prod 99.6%

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

      4. *-un-lft-identity 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. add-sqr-sqrt 99.4%

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

      7. fabs-sqr 99.4%

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

      8. add-sqr-sqrt 99.6%

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

      9. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. +-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.7%

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

Alternative 5: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.03999999910593033)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.0010000000474974513)
       (copysign x x)
       (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.03999999910593033f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.0010000000474974513f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03999999910593033))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.0010000000474974513))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(abs(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(-0.03999999910593033))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.0010000000474974513))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((abs(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) < -0.0399999991

   1. Initial program 49.8%

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

      1. +-commutative 49.8%

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

      2. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 11.3%

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

      2. frac-2neg 11.3%

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

      3. log-div 11.3%

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

   5. Applied egg-rr 15.7%

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

      1. sub-neg 15.7%

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

      2. sub-neg 15.7%

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

      3. fma-udef 15.7%

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

      4. unpow2 15.7%

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

      5. associate--r+ 46.4%

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

      6. +-inverses 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

      10. neg-sub0 99.8%

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

      11. sub-neg 99.8%

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

      12. +-commutative 99.8%

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

      13. distribute-neg-in 99.8%

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

      14. remove-double-neg 99.8%

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

      15. sub-neg 99.8%

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

   7. Simplified 99.8%

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

   if -0.0399999991 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.00100000005

   1. Initial program 19.8%

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

      1. +-commutative 19.8%

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

      2. hypot-1-def 19.8%

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

   3. Simplified 19.8%

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

   4. Taylor expanded in x around 0 19.3%

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

      1. log1p-def 96.4%

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

      2. rem-square-sqrt 41.0%

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

      3. fabs-sqr 41.0%

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

      4. rem-square-sqrt 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in x around 0 99.8%

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

   if 0.00100000005 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 66.4%

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

      1. +-commutative 66.4%

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

      2. hypot-1-def 99.0%

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

   3. Simplified 99.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 6: 98.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (+ (- x x) (/ -0.5 x))) x)
   (if (<= x 0.05000000074505806)
     (copysign (* 0.3333333333333333 (+ (* -0.5 (pow x 3.0)) (* x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((x - x) + (-0.5f / x))), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((0.3333333333333333f * ((-0.5f * powf(x, 3.0f)) + (x * 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(x - x) + Float32(Float32(-0.5) / x))), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(Float32(0.3333333333333333) * Float32(Float32(Float32(-0.5) * (x ^ Float32(3.0))) + 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)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log(((x - x) + (single(-0.5) / x))));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((single(0.3333333333333333) * ((single(-0.5) * (x ^ single(3.0))) + (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 x < -2

   1. Initial program 47.3%

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

      1. +-commutative 47.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 99.3%

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

      1. sub-neg 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

      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 98.1%

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

      7. associate-*r/ 98.1%

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

      8. metadata-eval 98.1%

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

      9. distribute-neg-frac 98.1%

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

      10. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

   if -2 < x < 0.0500000007

   1. Initial program 23.3%

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

      1. +-commutative 23.3%

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

      2. hypot-1-def 23.4%

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

   3. Simplified 23.4%

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

      1. add-cbrt-cube 23.3%

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

      2. pow1/3 23.4%

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

      3. log-pow 23.4%

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

      4. pow3 23.3%

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

      5. log-pow 23.3%

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

      6. *-un-lft-identity 23.3%

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

      7. *-un-lft-identity 23.3%

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

      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 23.5%

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

   5. Applied egg-rr 23.5%

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

   6. Taylor expanded in x around 0 98.3%

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

   if 0.0500000007 < x

   1. Initial program 65.3%

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

      1. +-commutative 65.3%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. *-un-lft-identity 99.9%

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

      6. add-sqr-sqrt 99.6%

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

      7. fabs-sqr 99.6%

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

      8. add-sqr-sqrt 99.9%

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

      9. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. +-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 98.6%

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

Alternative 7: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.03999999910593033)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.0010000000474974513)
     (copysign x x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.03999999910593033f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.0010000000474974513f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.03999999910593033))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.0010000000474974513))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.03999999910593033))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.0010000000474974513))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0399999991

   1. Initial program 49.8%

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

      1. +-commutative 49.8%

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

      2. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 11.3%

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

      2. frac-2neg 11.3%

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

      3. log-div 11.3%

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

   5. Applied egg-rr 15.7%

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

      1. sub-neg 15.7%

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

      2. sub-neg 15.7%

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

      3. fma-udef 15.7%

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

      4. unpow2 15.7%

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

      5. associate--r+ 46.4%

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

      6. +-inverses 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

      10. neg-sub0 99.8%

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

      11. sub-neg 99.8%

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

      12. +-commutative 99.8%

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

      13. distribute-neg-in 99.8%

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

      14. remove-double-neg 99.8%

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

      15. sub-neg 99.8%

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

   7. Simplified 99.8%

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

   if -0.0399999991 < x < 0.00100000005

   1. Initial program 19.8%

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

      1. +-commutative 19.8%

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

      2. hypot-1-def 19.8%

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

   3. Simplified 19.8%

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

   4. Taylor expanded in x around 0 19.3%

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

      1. log1p-def 96.4%

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

      2. rem-square-sqrt 41.0%

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

      3. fabs-sqr 41.0%

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

      4. rem-square-sqrt 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in x around 0 99.8%

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

   if 0.00100000005 < x

   1. Initial program 66.4%

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

      1. +-commutative 66.4%

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

      2. hypot-1-def 99.0%

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

   3. Simplified 99.0%

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

      1. *-un-lft-identity 99.0%

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

      2. *-commutative 99.0%

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

      3. log-prod 99.0%

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

      4. *-un-lft-identity 99.0%

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

      5. *-un-lft-identity 99.0%

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

      6. add-sqr-sqrt 98.8%

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

      7. fabs-sqr 98.8%

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

      8. add-sqr-sqrt 99.0%

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

      9. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. +-rgt-identity 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 8: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 47.3%

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

      1. +-commutative 47.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 99.3%

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

      1. sub-neg 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

      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 98.1%

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

      7. associate-*r/ 98.1%

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

      8. metadata-eval 98.1%

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

      9. distribute-neg-frac 98.1%

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

      10. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

   if -2 < x < 1

   1. Initial program 24.9%

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

      1. +-commutative 24.9%

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

      2. hypot-1-def 25.0%

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

   3. Simplified 25.0%

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

      1. add-cbrt-cube 24.9%

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

      2. pow1/3 25.0%

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

      3. log-pow 25.0%

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

      4. pow3 24.9%

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

      5. log-pow 25.0%

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

      6. *-un-lft-identity 25.0%

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

      7. *-un-lft-identity 25.0%

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

      8. add-sqr-sqrt 13.2%

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

      9. fabs-sqr 13.2%

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

      10. add-sqr-sqrt 25.1%

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

   5. Applied egg-rr 25.1%

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

   6. Taylor expanded in x around 0 97.3%

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

      1. *-commutative 97.3%

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

   8. Simplified 97.3%

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

   if 1 < x

   1. Initial program 63.8%

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

      1. +-commutative 63.8%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.6%

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

      1. rem-square-sqrt 99.4%

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

      2. fabs-sqr 99.4%

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

      3. rem-square-sqrt 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 98.0%

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

Alternative 9: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 47.3%

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

      1. +-commutative 47.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 99.3%

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

      1. sub-neg 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

      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 98.1%

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

      7. associate-*r/ 98.1%

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

      8. metadata-eval 98.1%

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

      9. distribute-neg-frac 98.1%

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

      10. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

   if -2 < x < 1

   1. Initial program 24.9%

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

      1. +-commutative 24.9%

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

      2. hypot-1-def 25.0%

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

   3. Simplified 25.0%

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

      1. add-cbrt-cube 24.9%

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

      2. pow1/3 25.0%

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

      3. log-pow 25.0%

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

      4. pow3 24.9%

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

      5. log-pow 25.0%

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

      6. *-un-lft-identity 25.0%

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

      7. *-un-lft-identity 25.0%

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

      8. add-sqr-sqrt 13.2%

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

      9. fabs-sqr 13.2%

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

      10. add-sqr-sqrt 25.1%

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

   5. Applied egg-rr 25.1%

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

   6. Taylor expanded in x around 0 97.3%

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

   if 1 < x

   1. Initial program 63.8%

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

      1. +-commutative 63.8%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.6%

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

      1. rem-square-sqrt 99.4%

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

      2. fabs-sqr 99.4%

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

      3. rem-square-sqrt 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 98.0%

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

Alternative 10: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -20.0)
   (copysign (log (- x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (+ (+ x x) (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -20.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(((x + x) + (0.5f / x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-20.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(x + x) + Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-20.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(((x + x) + (single(0.5) / x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -20

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 44.9%

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

      1. mul-1-neg 44.9%

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

   6. Simplified 44.9%

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

   if -20 < x < 1

   1. Initial program 25.4%

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

      1. +-commutative 25.4%

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

      2. hypot-1-def 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in x around 0 21.0%

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

      1. log1p-def 92.0%

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

      2. rem-square-sqrt 39.9%

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

      3. fabs-sqr 39.9%

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

      4. rem-square-sqrt 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in x around 0 96.2%

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

   if 1 < x

   1. Initial program 63.8%

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

      1. +-commutative 63.8%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.6%

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

      1. rem-square-sqrt 99.4%

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

      2. fabs-sqr 99.4%

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

      3. rem-square-sqrt 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 11: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 47.3%

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

      1. +-commutative 47.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 99.3%

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

      1. sub-neg 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

      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 98.1%

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

      7. associate-*r/ 98.1%

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

      8. metadata-eval 98.1%

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

      9. distribute-neg-frac 98.1%

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

      10. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

   if -2 < x < 1

   1. Initial program 24.9%

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

      1. +-commutative 24.9%

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

      2. hypot-1-def 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in x around 0 20.9%

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

      1. log1p-def 92.4%

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

      2. rem-square-sqrt 40.2%

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

      3. fabs-sqr 40.2%

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

      4. rem-square-sqrt 92.3%

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

   6. Simplified 92.3%

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

   7. Taylor expanded in x around 0 96.7%

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

   if 1 < x

   1. Initial program 63.8%

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

      1. +-commutative 63.8%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.6%

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

      1. rem-square-sqrt 99.4%

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

      2. fabs-sqr 99.4%

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

      3. rem-square-sqrt 99.6%

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

      4. associate-+r+ 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 97.7%

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

Alternative 12: 84.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -20.0)
   (copysign (log (- x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -20.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-20.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-20.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -20

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 44.9%

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

      1. mul-1-neg 44.9%

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

   6. Simplified 44.9%

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

   if -20 < x < 1

   1. Initial program 25.4%

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

      1. +-commutative 25.4%

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

      2. hypot-1-def 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in x around 0 21.0%

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

      1. log1p-def 92.0%

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

      2. rem-square-sqrt 39.9%

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

      3. fabs-sqr 39.9%

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

      4. rem-square-sqrt 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in x around 0 96.2%

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

   if 1 < x

   1. Initial program 63.8%

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

      1. +-commutative 63.8%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 96.4%

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

      1. rem-square-sqrt 96.2%

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

      2. fabs-sqr 96.2%

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

      3. rem-square-sqrt 96.4%

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

   6. Simplified 96.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 13: 69.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < -2

   1. Initial program 47.3%

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

      1. +-commutative 47.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 44.7%

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

      1. mul-1-neg 44.7%

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

   6. Simplified 44.7%

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

   if -2 < x

   1. Initial program 36.9%

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

      1. +-commutative 36.9%

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

      2. hypot-1-def 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in x around 0 27.6%

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

      1. log1p-def 77.1%

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

      2. rem-square-sqrt 41.0%

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

      3. fabs-sqr 41.0%

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

      4. rem-square-sqrt 77.0%

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

   6. Simplified 77.0%

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

4. Final simplification 70.1%

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

Alternative 14: 62.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 31.3%

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

      1. +-commutative 31.3%

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

      2. hypot-1-def 46.2%

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

   3. Simplified 46.2%

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

   4. Taylor expanded in x around 0 27.7%

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

      1. log1p-def 78.9%

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

      2. rem-square-sqrt 28.8%

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

      3. fabs-sqr 28.8%

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

      4. rem-square-sqrt 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in x around 0 72.3%

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

   if 1 < x

   1. Initial program 63.8%

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

      1. +-commutative 63.8%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 42.7%

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

      1. log1p-def 42.7%

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

      2. rem-square-sqrt 42.7%

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

      3. fabs-sqr 42.7%

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

      4. rem-square-sqrt 42.7%

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

   6. Simplified 42.7%

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

4. Final simplification 65.1%

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

Alternative 15: 54.8% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (copysign x x))

C

float code(float x) {
	return copysignf(x, x);
}

Julia

function code(x)
	return copysign(x, x)
end

MATLAB

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

Derivation

1. Initial program 39.1%

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

   1. +-commutative 39.1%

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

   2. hypot-1-def 59.3%

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

3. Simplified 59.3%

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

4. Taylor expanded in x around 0 31.3%

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

   1. log1p-def 70.1%

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

   2. rem-square-sqrt 32.2%

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

   3. fabs-sqr 32.2%

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

   4. rem-square-sqrt 60.5%

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

6. Simplified 60.5%

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

7. Taylor expanded in x around 0 57.9%

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

8. Final simplification 57.9%

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

Developer target: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2023319 
(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))