Rust f32::asinh

Percentage Accurate: 37.4% → 99.1%

Time: 9.0s

Alternatives: 13

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: 37.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + 1\\ t_1 := {t\_0}^{2}\\ t_2 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_2 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + {x}^{2} \cdot \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(-0.041666666666666664, \frac{3}{t\_0} + \frac{3}{t\_1}, \left({x}^{2} \cdot 0.001388888888888889\right) \cdot \left(\frac{45}{t\_0} + \left(\frac{45}{t\_1} + \frac{30}{{t\_0}^{3}}\right)\right)\right), \frac{0.5}{t\_0}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ (fabs x) 1.0))
        (t_1 (pow t_0 2.0))
        (t_2 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_2 -1.0)
     (copysign (log (/ -0.5 x)) x)
     (if (<= t_2 0.05000000074505806)
       (copysign
        (+
         (log1p (fabs x))
         (*
          (pow x 2.0)
          (fma
           (pow x 2.0)
           (fma
            -0.041666666666666664
            (+ (/ 3.0 t_0) (/ 3.0 t_1))
            (*
             (* (pow x 2.0) 0.001388888888888889)
             (+ (/ 45.0 t_0) (+ (/ 45.0 t_1) (/ 30.0 (pow t_0 3.0))))))
           (/ 0.5 t_0))))
        x)
       (copysign (log1p (+ (hypot 1.0 x) (+ x -1.0))) x)))))

C

float code(float x) {
	float t_0 = fabsf(x) + 1.0f;
	float t_1 = powf(t_0, 2.0f);
	float t_2 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_2 <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (t_2 <= 0.05000000074505806f) {
		tmp = copysignf((log1pf(fabsf(x)) + (powf(x, 2.0f) * fmaf(powf(x, 2.0f), fmaf(-0.041666666666666664f, ((3.0f / t_0) + (3.0f / t_1)), ((powf(x, 2.0f) * 0.001388888888888889f) * ((45.0f / t_0) + ((45.0f / t_1) + (30.0f / powf(t_0, 3.0f)))))), (0.5f / t_0)))), x);
	} else {
		tmp = copysignf(log1pf((hypotf(1.0f, x) + (x + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(abs(x) + Float32(1.0))
	t_1 = t_0 ^ Float32(2.0)
	t_2 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_2 <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (t_2 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(log1p(abs(x)) + Float32((x ^ Float32(2.0)) * fma((x ^ Float32(2.0)), fma(Float32(-0.041666666666666664), Float32(Float32(Float32(3.0) / t_0) + Float32(Float32(3.0) / t_1)), Float32(Float32((x ^ Float32(2.0)) * Float32(0.001388888888888889)) * Float32(Float32(Float32(45.0) / t_0) + Float32(Float32(Float32(45.0) / t_1) + Float32(Float32(30.0) / (t_0 ^ Float32(3.0))))))), Float32(Float32(0.5) / t_0)))), x);
	else
		tmp = copysign(log1p(Float32(hypot(Float32(1.0), x) + Float32(x + Float32(-1.0)))), 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) #s(literal 1 binary32))))) x) < -1

   1. Initial program 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around -inf 96.1%

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

      1. mul-1-neg 96.1%

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

      2. *-commutative 96.1%

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

      3. distribute-rgt-neg-in 96.1%

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

      4. mul-1-neg 96.1%

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

      5. unsub-neg 96.1%

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

      6. sub-neg 96.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 9.5%

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

      10. associate-*r/ 9.5%

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

      11. metadata-eval 9.5%

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

      12. distribute-neg-frac 9.5%

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

      13. metadata-eval 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in x around 0 98.2%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 23.1%

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

      1. +-commutative 23.1%

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

      2. hypot-1-def 23.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in x around 0 24.2%

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

   7. Simplified 99.8%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 63.3%

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

      1. +-commutative 63.3%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

      1. add-cbrt-cube 50.2%

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

      2. pow1/3 49.5%

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

      3. log-pow 49.5%

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

      4. pow3 49.5%

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

      5. log-pow 97.2%

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

      6. add-sqr-sqrt 97.2%

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

      7. fabs-sqr 97.2%

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

      8. add-sqr-sqrt 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. log1p-expm1-u 97.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)}, x\right) \]

      2. log1p-undefine 97.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)}, x\right) \]

      3. expm1-undefine 97.2%

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

      4. associate-*r* 98.4%

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

      5. metadata-eval 98.4%

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

      6. *-un-lft-identity 98.4%

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

      7. add-exp-log 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. log1p-define 98.4%

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

      2. sub-neg 98.4%

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

      3. +-commutative 98.4%

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

      4. metadata-eval 98.4%

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

      5. associate-+l+ 98.5%

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

   10. Simplified 98.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + {x}^{2} \cdot \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(-0.041666666666666664, \frac{3}{\left|x\right| + 1} + \frac{3}{{\left(\left|x\right| + 1\right)}^{2}}, \left({x}^{2} \cdot 0.001388888888888889\right) \cdot \left(\frac{45}{\left|x\right| + 1} + \left(\frac{45}{{\left(\left|x\right| + 1\right)}^{2}} + \frac{30}{{\left(\left|x\right| + 1\right)}^{3}}\right)\right)\right), \frac{0.5}{\left|x\right| + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (/ -0.5 x)) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (*
         x
         (+ 1.0 (* (pow x 2.0) (- (* (pow x 2.0) 0.075) 0.16666666666666666))))
        x)
       (copysign (log1p (+ (hypot 1.0 x) (+ x -1.0))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((powf(x, 2.0f) * 0.075f) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(log1pf((hypotf(1.0f, x) + (x + -1.0f))), 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(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(0.075)) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(log1p(Float32(hypot(Float32(1.0), x) + Float32(x + Float32(-1.0)))), 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) #s(literal 1 binary32))))) x) < -1

   1. Initial program 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around -inf 96.1%

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

      1. mul-1-neg 96.1%

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

      2. *-commutative 96.1%

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

      3. distribute-rgt-neg-in 96.1%

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

      4. mul-1-neg 96.1%

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

      5. unsub-neg 96.1%

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

      6. sub-neg 96.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 9.5%

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

      10. associate-*r/ 9.5%

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

      11. metadata-eval 9.5%

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

      12. distribute-neg-frac 9.5%

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

      13. metadata-eval 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in x around 0 98.2%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 23.1%

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

      1. +-commutative 23.1%

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

      2. hypot-1-def 23.3%

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

   3. Simplified 23.3%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. 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 23.3%

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

      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 23.5%

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

      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. add-sqr-sqrt 11.0%

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

      7. fabs-sqr 11.0%

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

      8. add-sqr-sqrt 23.5%

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

   6. Applied egg-rr 23.5%

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

   7. Taylor expanded in x around 0 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)}, x\right) \]

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 63.3%

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

      1. +-commutative 63.3%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

      1. add-cbrt-cube 50.2%

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

      2. pow1/3 49.5%

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

      3. log-pow 49.5%

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

      4. pow3 49.5%

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

      5. log-pow 97.2%

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

      6. add-sqr-sqrt 97.2%

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

      7. fabs-sqr 97.2%

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

      8. add-sqr-sqrt 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. log1p-expm1-u 97.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)}, x\right) \]

      2. log1p-undefine 97.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)}, x\right) \]

      3. expm1-undefine 97.2%

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

      4. associate-*r* 98.4%

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

      5. metadata-eval 98.4%

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

      6. *-un-lft-identity 98.4%

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

      7. add-exp-log 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. log1p-define 98.4%

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

      2. sub-neg 98.4%

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

      3. +-commutative 98.4%

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

      4. metadata-eval 98.4%

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

      5. associate-+l+ 98.5%

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

   10. Simplified 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log1p (+ (hypot 1.0 x) (+ x -1.0))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(log1pf((hypotf(1.0f, x) + (x + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log1p(Float32(hypot(Float32(1.0), x) + Float32(x + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 51.0%

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

      1. +-commutative 51.0%

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

      2. hypot-1-def 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around -inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. *-commutative 96.6%

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

      3. distribute-rgt-neg-in 96.6%

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

      4. mul-1-neg 96.6%

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

      5. unsub-neg 96.6%

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

      6. sub-neg 96.6%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 8.9%

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

      10. associate-*r/ 8.9%

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

      11. metadata-eval 8.9%

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

      12. distribute-neg-frac 8.9%

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

      13. metadata-eval 8.9%

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

   7. Simplified 8.9%

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

   8. Taylor expanded in x around 0 99.1%

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

   if -5 < x < 0.0500000007

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. hypot-1-def 23.9%

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

   3. Simplified 23.9%

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

      1. add-cbrt-cube 23.9%

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

      2. pow1/3 23.9%

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

      3. log-pow 24.0%

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

      4. pow3 24.1%

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

      5. log-pow 23.9%

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

      6. add-sqr-sqrt 10.9%

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

      7. fabs-sqr 10.9%

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

      8. add-sqr-sqrt 24.1%

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

   6. Applied egg-rr 24.1%

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

   7. Taylor expanded in x around 0 98.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   8. Step-by-step derivation

      1. distribute-lft-in 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot 1 + x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]

      2. *-rgt-identity 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right), x\right) \]

      3. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left(x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

      4. associate-*r* 99.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.16666666666666666}, x\right) \]

      5. unpow2 99.0%

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

      6. cube-mult 99.0%

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

   9. Simplified 99.0%

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

   if 0.0500000007 < x

   1. Initial program 63.3%

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

      1. +-commutative 63.3%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

      1. add-cbrt-cube 50.2%

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

      2. pow1/3 49.5%

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

      3. log-pow 49.5%

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

      4. pow3 49.5%

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

      5. log-pow 97.2%

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

      6. add-sqr-sqrt 97.2%

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

      7. fabs-sqr 97.2%

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

      8. add-sqr-sqrt 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. log1p-expm1-u 97.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)}, x\right) \]

      2. log1p-undefine 97.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)}, x\right) \]

      3. expm1-undefine 97.2%

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

      4. associate-*r* 98.4%

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

      5. metadata-eval 98.4%

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

      6. *-un-lft-identity 98.4%

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

      7. add-exp-log 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. log1p-define 98.4%

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

      2. sub-neg 98.4%

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

      3. +-commutative 98.4%

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

      4. metadata-eval 98.4%

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

      5. associate-+l+ 98.5%

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

   10. Simplified 98.5%

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

Alternative 4: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 51.0%

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

      1. +-commutative 51.0%

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

      2. hypot-1-def 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around -inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. *-commutative 96.6%

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

      3. distribute-rgt-neg-in 96.6%

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

      4. mul-1-neg 96.6%

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

      5. unsub-neg 96.6%

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

      6. sub-neg 96.6%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 8.9%

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

      10. associate-*r/ 8.9%

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

      11. metadata-eval 8.9%

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

      12. distribute-neg-frac 8.9%

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

      13. metadata-eval 8.9%

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

   7. Simplified 8.9%

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

   8. Taylor expanded in x around 0 99.1%

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

   if -5 < x < 0.0500000007

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. hypot-1-def 23.9%

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

   3. Simplified 23.9%

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

      1. add-cbrt-cube 23.9%

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

      2. pow1/3 23.9%

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

      3. log-pow 24.0%

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

      4. pow3 24.1%

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

      5. log-pow 23.9%

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

      6. add-sqr-sqrt 10.9%

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

      7. fabs-sqr 10.9%

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

      8. add-sqr-sqrt 24.1%

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

   6. Applied egg-rr 24.1%

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

   7. Taylor expanded in x around 0 98.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   8. Step-by-step derivation

      1. distribute-lft-in 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot 1 + x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]

      2. *-rgt-identity 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right), x\right) \]

      3. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left(x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

      4. associate-*r* 99.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.16666666666666666}, x\right) \]

      5. unpow2 99.0%

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

      6. cube-mult 99.0%

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

   9. Simplified 99.0%

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

   if 0.0500000007 < x

   1. Initial program 63.3%

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

      1. +-commutative 63.3%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around 0 63.3%

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

      1. rem-square-sqrt 63.3%

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

      2. fabs-sqr 63.3%

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

      3. metadata-eval 63.3%

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

      4. unpow2 63.3%

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

      5. hypot-undefine 98.4%

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

      6. rem-square-sqrt 98.4%

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

   7. Simplified 98.4%

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

Alternative 5: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.5)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.5))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 51.0%

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

      1. +-commutative 51.0%

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

      2. hypot-1-def 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around -inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. *-commutative 96.6%

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

      3. distribute-rgt-neg-in 96.6%

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

      4. mul-1-neg 96.6%

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

      5. unsub-neg 96.6%

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

      6. sub-neg 96.6%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 8.9%

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

      10. associate-*r/ 8.9%

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

      11. metadata-eval 8.9%

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

      12. distribute-neg-frac 8.9%

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

      13. metadata-eval 8.9%

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

   7. Simplified 8.9%

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

   8. Taylor expanded in x around 0 99.1%

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

   if -5 < x < 0.5

   1. Initial program 25.9%

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

      1. +-commutative 25.9%

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

      2. hypot-1-def 26.1%

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

   3. Simplified 26.1%

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

      1. add-cbrt-cube 26.1%

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

      2. pow1/3 26.1%

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

      3. log-pow 26.2%

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

      4. pow3 26.3%

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

      5. log-pow 26.1%

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

      6. add-sqr-sqrt 13.5%

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

      7. fabs-sqr 13.5%

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

      8. add-sqr-sqrt 26.3%

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

   6. Applied egg-rr 26.3%

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

   7. Taylor expanded in x around 0 97.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   8. Step-by-step derivation

      1. distribute-lft-in 97.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot 1 + x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]

      2. *-rgt-identity 97.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right), x\right) \]

      3. *-commutative 97.8%

         \[\leadsto \mathsf{copysign}\left(x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

      4. associate-*r* 97.8%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.16666666666666666}, x\right) \]

      5. unpow2 97.8%

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

      6. cube-mult 97.8%

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

   9. Simplified 97.8%

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

   if 0.5 < x

   1. Initial program 61.0%

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

      1. +-commutative 61.0%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around -inf 5.4%

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

      1. mul-1-neg 5.4%

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

      2. *-commutative 5.4%

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

      3. distribute-rgt-neg-in 5.4%

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

      4. mul-1-neg 5.4%

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

      5. unsub-neg 5.4%

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

      6. sub-neg 5.4%

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

      7. rem-square-sqrt 9.2%

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

      8. fabs-sqr 9.2%

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

      9. rem-square-sqrt 5.4%

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

      10. associate-*r/ 5.4%

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

      11. metadata-eval 5.4%

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

      12. distribute-neg-frac 5.4%

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

      13. metadata-eval 5.4%

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

   7. Simplified 5.4%

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

      1. metadata-eval 5.4%

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

      2. add-sqr-sqrt 5.4%

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

      3. sqrt-unprod 5.4%

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

      4. sqr-neg 5.4%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 13.9%

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

      7. frac-2neg 13.9%

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

      8. div-inv 13.9%

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

      9. metadata-eval 13.9%

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

      10. cancel-sign-sub-inv 13.9%

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

      11. div-inv 13.9%

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

   9. Applied egg-rr 13.9%

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

   10. Taylor expanded in x around 0 95.7%

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

Alternative 6: 97.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around -inf 96.1%

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

      1. mul-1-neg 96.1%

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

      2. *-commutative 96.1%

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

      3. distribute-rgt-neg-in 96.1%

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

      4. mul-1-neg 96.1%

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

      5. unsub-neg 96.1%

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

      6. sub-neg 96.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 9.5%

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

      10. associate-*r/ 9.5%

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

      11. metadata-eval 9.5%

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

      12. distribute-neg-frac 9.5%

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

      13. metadata-eval 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in x around 0 98.2%

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

   if -2 < x < 0.5

   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. Add Preprocessing

   5. Taylor expanded in x around 0 21.0%

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

      1. log1p-define 91.3%

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

      2. rem-square-sqrt 46.9%

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

      3. fabs-sqr 46.9%

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

      4. rem-square-sqrt 91.2%

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

   7. Simplified 91.2%

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

   8. Taylor expanded in x around 0 96.4%

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

   if 0.5 < x

   1. Initial program 61.0%

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

      1. +-commutative 61.0%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around -inf 5.4%

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

      1. mul-1-neg 5.4%

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

      2. *-commutative 5.4%

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

      3. distribute-rgt-neg-in 5.4%

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

      4. mul-1-neg 5.4%

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

      5. unsub-neg 5.4%

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

      6. sub-neg 5.4%

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

      7. rem-square-sqrt 9.2%

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

      8. fabs-sqr 9.2%

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

      9. rem-square-sqrt 5.4%

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

      10. associate-*r/ 5.4%

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

      11. metadata-eval 5.4%

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

      12. distribute-neg-frac 5.4%

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

      13. metadata-eval 5.4%

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

   7. Simplified 5.4%

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

      1. metadata-eval 5.4%

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

      2. add-sqr-sqrt 5.4%

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

      3. sqrt-unprod 5.4%

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

      4. sqr-neg 5.4%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 13.9%

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

      7. frac-2neg 13.9%

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

      8. div-inv 13.9%

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

      9. metadata-eval 13.9%

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

      10. cancel-sign-sub-inv 13.9%

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

      11. div-inv 13.9%

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

   9. Applied egg-rr 13.9%

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

   10. Taylor expanded in x around 0 95.7%

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

Alternative 7: 97.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around -inf 96.1%

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

      1. mul-1-neg 96.1%

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

      2. *-commutative 96.1%

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

      3. distribute-rgt-neg-in 96.1%

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

      4. mul-1-neg 96.1%

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

      5. unsub-neg 96.1%

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

      6. sub-neg 96.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 9.5%

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

      10. associate-*r/ 9.5%

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

      11. metadata-eval 9.5%

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

      12. distribute-neg-frac 9.5%

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

      13. metadata-eval 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in x around 0 98.2%

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

   if -2 < x < 0.5

   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. Add Preprocessing

   5. Taylor expanded in x around 0 21.0%

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

      1. log1p-define 91.3%

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

      2. rem-square-sqrt 46.9%

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

      3. fabs-sqr 46.9%

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

      4. rem-square-sqrt 91.2%

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

   7. Simplified 91.2%

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

   8. Taylor expanded in x around 0 96.4%

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

   if 0.5 < x

   1. Initial program 61.0%

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

      1. +-commutative 61.0%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around inf 94.2%

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

      1. +-commutative 94.2%

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

      2. rem-square-sqrt 94.2%

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

      3. fabs-sqr 94.2%

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

      4. rem-square-sqrt 94.2%

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

      5. *-inverses 94.2%

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

      6. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

Alternative 8: 97.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around -inf 96.1%

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

      1. mul-1-neg 96.1%

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

      2. *-commutative 96.1%

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

      3. distribute-rgt-neg-in 96.1%

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

      4. mul-1-neg 96.1%

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

      5. unsub-neg 96.1%

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

      6. sub-neg 96.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 9.5%

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

      10. associate-*r/ 9.5%

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

      11. metadata-eval 9.5%

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

      12. distribute-neg-frac 9.5%

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

      13. metadata-eval 9.5%

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

   7. Simplified 9.5%

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

      1. metadata-eval 9.5%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 6.0%

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

      4. sqr-neg 6.0%

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

      5. sqrt-unprod 6.0%

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

      6. add-sqr-sqrt 6.0%

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

      7. frac-2neg 6.0%

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

      8. div-inv 6.0%

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

      9. metadata-eval 6.0%

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

      10. cancel-sign-sub-inv 6.0%

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

      11. div-inv 6.0%

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

   9. Applied egg-rr 6.0%

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

   10. Taylor expanded in x around 0 -0.0%

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

       1. add-sqr-sqrt -0.0%

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

       2. sqrt-unprod 51.9%

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

       3. frac-times 50.0%

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

       4. metadata-eval 50.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\sqrt{\frac{\color{blue}{0.25}}{x \cdot x}}\right), x\right) \]

       5. metadata-eval 50.0%

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

       6. frac-times 51.9%

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

       7. sqrt-unprod 98.2%

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

       8. add-sqr-sqrt 98.2%

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

       9. frac-2neg 98.2%

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

       10. metadata-eval 98.2%

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

       11. div-inv 98.2%

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

   12. Applied egg-rr 98.2%

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

       1. metadata-eval 98.2%

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

       2. distribute-lft-neg-in 98.2%

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

       3. *-commutative 98.2%

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

       4. associate-*l/ 98.2%

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

       5. metadata-eval 98.2%

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

       6. distribute-frac-neg2 98.2%

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

       7. distribute-neg-frac 98.2%

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

       8. metadata-eval 98.2%

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

       9. rem-exp-log -0.0%

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

       10. rem-square-sqrt -0.0%

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

       11. fabs-sqr -0.0%

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

       12. rem-square-sqrt -0.0%

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

       13. fabs-neg -0.0%

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

       14. log-rec -0.0%

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

       15. associate-/r/ -0.0%

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

       16. metadata-eval -0.0%

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

       17. *-commutative -0.0%

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

       18. rem-square-sqrt -0.0%

           \[\leadsto \mathsf{copysign}\left(\log \left(-e^{\left|\color{blue}{\sqrt{\log \left(x \cdot 2\right)} \cdot \sqrt{\log \left(x \cdot 2\right)}}\right|}\right), x\right) \]

       19. fabs-sqr -0.0%

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

       20. rem-square-sqrt -0.0%

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

       21. rem-exp-log 95.3%

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

   14. Simplified 95.3%

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

   if -2 < x < 0.5

   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. Add Preprocessing

   5. Taylor expanded in x around 0 21.0%

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

      1. log1p-define 91.3%

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

      2. rem-square-sqrt 46.9%

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

      3. fabs-sqr 46.9%

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

      4. rem-square-sqrt 91.2%

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

   7. Simplified 91.2%

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

   8. Taylor expanded in x around 0 96.4%

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

   if 0.5 < x

   1. Initial program 61.0%

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

      1. +-commutative 61.0%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around inf 94.2%

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

      1. +-commutative 94.2%

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

      2. rem-square-sqrt 94.2%

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

      3. fabs-sqr 94.2%

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

      4. rem-square-sqrt 94.2%

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

      5. *-inverses 94.2%

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

      6. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

Alternative 9: 82.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((x * -2.0f)), 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 * Float32(-2.0))), 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 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around -inf 96.1%

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

      1. mul-1-neg 96.1%

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

      2. *-commutative 96.1%

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

      3. distribute-rgt-neg-in 96.1%

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

      4. mul-1-neg 96.1%

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

      5. unsub-neg 96.1%

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

      6. sub-neg 96.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 9.5%

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

      10. associate-*r/ 9.5%

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

      11. metadata-eval 9.5%

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

      12. distribute-neg-frac 9.5%

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

      13. metadata-eval 9.5%

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

   7. Simplified 9.5%

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

      1. metadata-eval 9.5%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 6.0%

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

      4. sqr-neg 6.0%

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

      5. sqrt-unprod 6.0%

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

      6. add-sqr-sqrt 6.0%

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

      7. frac-2neg 6.0%

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

      8. div-inv 6.0%

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

      9. metadata-eval 6.0%

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

      10. cancel-sign-sub-inv 6.0%

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

      11. div-inv 6.0%

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

   9. Applied egg-rr 6.0%

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

   10. Taylor expanded in x around 0 -0.0%

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

       1. add-sqr-sqrt -0.0%

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

       2. sqrt-unprod 51.9%

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

       3. frac-times 50.0%

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

       4. metadata-eval 50.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\sqrt{\frac{\color{blue}{0.25}}{x \cdot x}}\right), x\right) \]

       5. metadata-eval 50.0%

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

       6. frac-times 51.9%

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

       7. sqrt-unprod 98.2%

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

       8. add-sqr-sqrt 98.2%

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

       9. frac-2neg 98.2%

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

       10. metadata-eval 98.2%

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

       11. div-inv 98.2%

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

   12. Applied egg-rr 98.2%

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

       1. metadata-eval 98.2%

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

       2. distribute-lft-neg-in 98.2%

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

       3. *-commutative 98.2%

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

       4. associate-*l/ 98.2%

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

       5. metadata-eval 98.2%

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

       6. distribute-frac-neg2 98.2%

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

       7. distribute-neg-frac 98.2%

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

       8. metadata-eval 98.2%

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

       9. rem-exp-log -0.0%

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

       10. rem-square-sqrt -0.0%

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

       11. fabs-sqr -0.0%

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

       12. rem-square-sqrt -0.0%

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

       13. fabs-neg -0.0%

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

       14. log-rec -0.0%

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

       15. associate-/r/ -0.0%

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

       16. metadata-eval -0.0%

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

       17. *-commutative -0.0%

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

       18. rem-square-sqrt -0.0%

           \[\leadsto \mathsf{copysign}\left(\log \left(-e^{\left|\color{blue}{\sqrt{\log \left(x \cdot 2\right)} \cdot \sqrt{\log \left(x \cdot 2\right)}}\right|}\right), x\right) \]

       19. fabs-sqr -0.0%

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

       20. rem-square-sqrt -0.0%

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

       21. rem-exp-log 95.3%

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

   14. Simplified 95.3%

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

   if -2 < x

   1. Initial program 36.5%

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

      1. +-commutative 36.5%

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

      2. hypot-1-def 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in x around 0 28.0%

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

      1. log1p-define 76.3%

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

      2. rem-square-sqrt 45.8%

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

      3. fabs-sqr 45.8%

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

      4. rem-square-sqrt 76.3%

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

   7. Simplified 76.3%

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

Alternative 10: 68.9% 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 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around -inf 44.3%

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

      1. mul-1-neg 44.3%

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

   7. Simplified 44.3%

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

   if -2 < x

   1. Initial program 36.5%

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

      1. +-commutative 36.5%

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

      2. hypot-1-def 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in x around 0 28.0%

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

      1. log1p-define 76.3%

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

      2. rem-square-sqrt 45.8%

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

      3. fabs-sqr 45.8%

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

      4. rem-square-sqrt 76.3%

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

   7. Simplified 76.3%

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

Alternative 11: 62.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 34.3%

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

      1. +-commutative 34.3%

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

      2. hypot-1-def 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in x around 0 28.6%

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

      1. log1p-define 75.7%

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

      2. rem-square-sqrt 31.6%

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

      3. fabs-sqr 31.6%

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

      4. rem-square-sqrt 61.3%

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

   7. Simplified 61.3%

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

   8. Taylor expanded in x around 0 68.4%

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

   if 2 < x

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around 0 43.5%

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

      1. log1p-define 43.5%

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

      2. rem-square-sqrt 43.5%

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

      3. fabs-sqr 43.5%

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

      4. rem-square-sqrt 43.5%

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

   7. Simplified 43.5%

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

Alternative 12: 54.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 40.3%

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

   1. +-commutative 40.3%

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

   2. hypot-1-def 60.5%

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

3. Simplified 60.5%

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

5. Taylor expanded in x around 0 32.1%

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

   1. log1p-define 68.3%

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

   2. rem-square-sqrt 34.3%

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

   3. fabs-sqr 34.3%

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

   4. rem-square-sqrt 57.2%

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

7. Simplified 57.2%

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

8. Taylor expanded in x around 0 55.5%

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

Alternative 13: 15.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(-1, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign -1.0 x))

C

float code(float x) {
	return copysignf(-1.0f, x);
}

Julia

function code(x)
	return copysign(Float32(-1.0), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(-1.0));
end

Derivation

1. Initial program 40.3%

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

   1. +-commutative 40.3%

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

   2. hypot-1-def 60.5%

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

3. Simplified 60.5%

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

5. Taylor expanded in x around -inf 14.7%

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

   1. distribute-lft-out 14.7%

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

7. Simplified 14.7%

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

8. Taylor expanded in x around 0 16.3%

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

   1. rem-square-sqrt 8.2%

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

   2. fabs-sqr 8.2%

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

   3. rem-square-sqrt 16.3%

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

   4. *-inverses 16.3%

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

   5. metadata-eval 16.3%

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

10. Simplified 16.3%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{-1}, x\right) \]
11. Add Preprocessing

Developer Target 1: 99.7% 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 2024157 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (! :herbie-platform default (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

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