Rust f32::asinh

Percentage Accurate: 37.6% → 99.6%

Time: 13.5s

Alternatives: 14

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: 37.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.5 \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\mathsf{copysign}\left(\langle \left( \log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right) \right)_{\text{binary64}} \rangle_{\text{binary32}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(x \cdot \left(\frac{x}{\left|x\right| + 1} \cdot 0.5\right) + \mathsf{log1p}\left(-x \cdot x\right)\right) - \mathsf{log1p}\left(-\left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (or (<= t_0 -0.5) (not (<= t_0 0.0)))
     (copysign
      (cast (! :precision binary64 (log (+ (fabs x) (hypot 1.0 x)))))
      x)
     (copysign
      (-
       (+ (* x (* (/ x (+ (fabs x) 1.0)) 0.5)) (log1p (- (* x x))))
       (log1p (- (fabs x))))
      x))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp_1;
	if ((t_0 <= -0.5f) || !(t_0 <= 0.0f)) {
		double tmp_2 = log((fabs(x) + hypot(1.0, x)));
		tmp_1 = copysignf(((float) tmp_2), x);
	} else {
		tmp_1 = copysignf((((x * ((x / (fabsf(x) + 1.0f)) * 0.5f)) + log1pf(-(x * x))) - log1pf(-fabsf(x))), x);
	}
	return tmp_1;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp_1 = Float32(0.0)
	if ((t_0 <= Float32(-0.5)) || !(t_0 <= Float32(0.0)))
		tmp_2 = log(Float64(abs(x) + hypot(1.0, x)))
		tmp_1 = copysign(Float32(tmp_2), x);
	else
		tmp_1 = copysign(Float32(Float32(Float32(x * Float32(Float32(x / Float32(abs(x) + Float32(1.0))) * Float32(0.5))) + log1p(Float32(-Float32(x * x)))) - log1p(Float32(-abs(x)))), x);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -0.5 or 0.0 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 55.3%

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

      1. +-commutative 55.3%

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

      2. hypot-1-def 96.7%

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

   3. Simplified 96.7%

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

      1. rewrite-binary32/binary64 100.0%

         \[\leadsto \color{blue}{\mathsf{copysign}\left(\langle \log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right) \rangle_{\text{binary64}}, x\right)} \]

   5. Applied rewrite-once 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\langle \color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \rangle_{\text{binary64}}}, x\right) \]

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.0

   1. Initial program 18.5%

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

      1. +-commutative 18.5%

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

      2. hypot-1-def 18.7%

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

   3. Simplified 18.7%

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

   4. Taylor expanded in x around 0 19.7%

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

      1. log1p-def 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-udef 19.7%

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

      3. flip-+ 19.7%

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

      4. log-div 20.1%

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

      5. associate-+r- 20.1%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot \left(\frac{x}{\left|x\right| + 1} \cdot 0.5\right) + \mathsf{log1p}\left(x \cdot \left(-x\right)\right)\right) - \mathsf{log1p}\left(-\left|x\right|\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5 \lor \neg \left(\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0\right):\\ \;\;\;\;\mathsf{copysign}\left(\langle \log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right) \rangle_{\text{binary64}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(x \cdot \left(\frac{x}{\left|x\right| + 1} \cdot 0.5\right) + \mathsf{log1p}\left(-x \cdot x\right)\right) - \mathsf{log1p}\left(-\left|x\right|\right), x\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + 1\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_1 \leq -0.5 \lor \neg \left(t_1 \leq 0.009999999776482582\right):\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{t_0} + \frac{3}{{t_0}^{2}}\right), 0.5 \cdot \frac{x \cdot x}{t_0}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ (fabs x) 1.0))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (or (<= t_1 -0.5) (not (<= t_1 0.009999999776482582)))
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (copysign
      (+
       (log1p (fabs x))
       (fma
        -0.041666666666666664
        (* (pow x 4.0) (+ (/ 3.0 t_0) (/ 3.0 (pow t_0 2.0))))
        (* 0.5 (/ (* x x) t_0))))
      x))))

C

float code(float x) {
	float t_0 = fabsf(x) + 1.0f;
	float t_1 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if ((t_1 <= -0.5f) || !(t_1 <= 0.009999999776482582f)) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else {
		tmp = copysignf((log1pf(fabsf(x)) + fmaf(-0.041666666666666664f, (powf(x, 4.0f) * ((3.0f / t_0) + (3.0f / powf(t_0, 2.0f)))), (0.5f * ((x * x) / t_0)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(abs(x) + Float32(1.0))
	t_1 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if ((t_1 <= Float32(-0.5)) || !(t_1 <= Float32(0.009999999776482582)))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	else
		tmp = copysign(Float32(log1p(abs(x)) + fma(Float32(-0.041666666666666664), Float32((x ^ Float32(4.0)) * Float32(Float32(Float32(3.0) / t_0) + Float32(Float32(3.0) / (t_0 ^ Float32(2.0))))), Float32(Float32(0.5) * Float32(Float32(x * x) / t_0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -0.5 or 0.00999999978 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 54.7%

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

      1. +-commutative 54.7%

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

      2. hypot-1-def 99.1%

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

   3. Simplified 99.1%

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

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.00999999978

   1. Initial program 21.5%

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

      1. +-commutative 21.5%

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

      2. hypot-1-def 21.7%

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

   3. Simplified 21.7%

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

   4. Taylor expanded in x around 0 22.7%

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

      1. log1p-def 100.0%

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

      2. fma-def 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. unpow2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5 \lor \neg \left(\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.009999999776482582\right):\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \mathsf{fma}\left(-0.041666666666666664, {x}^{4} \cdot \left(\frac{3}{\left|x\right| + 1} + \frac{3}{{\left(\left|x\right| + 1\right)}^{2}}\right), 0.5 \cdot \frac{x \cdot x}{\left|x\right| + 1}\right), x\right)\\ \end{array} \]

Alternative 3: 99.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (or (<= t_0 -0.5) (not (<= t_0 0.009999999776482582)))
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (copysign (+ (/ (* (* x x) 0.5) (+ x 1.0)) (log1p x)) x))))

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -0.5 or 0.00999999978 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 54.7%

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

      1. +-commutative 54.7%

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

      2. hypot-1-def 99.1%

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

   3. Simplified 99.1%

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

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.00999999978

   1. Initial program 21.5%

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

      1. +-commutative 21.5%

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

      2. hypot-1-def 21.7%

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

   3. Simplified 21.7%

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

   4. Taylor expanded in x around 0 22.7%

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

      1. log1p-def 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. add-sqr-sqrt_binary32 98.4%

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

   8. Applied rewrite-once 98.4%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{0.5 \cdot \left(x \cdot x\right)}{x + 1} + \mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

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

Alternative 4: 97.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 55.8%

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

      1. +-commutative 55.8%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. unsub-neg 96.5%

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

   6. Simplified 96.5%

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

   if -0.5 < x < 1

   1. Initial program 22.1%

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

      1. +-commutative 22.1%

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

      2. hypot-1-def 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in x around 0 22.8%

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

      1. log1p-def 99.5%

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

      2. unpow2 99.5%

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

   6. Simplified 99.5%

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

      1. add-sqr-sqrt_binary32 98.0%

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

   8. Applied rewrite-once 98.0%

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

   10. Applied egg-rr 99.5%

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

   if 1 < x

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-1-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. associate-*r/ 97.6%

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

      2. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

4. Final simplification 98.3%

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

Alternative 5: 98.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 55.8%

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

      1. +-commutative 55.8%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 97.3%

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

      1. cancel-sign-sub-inv 97.3%

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

      2. mul-1-neg 97.3%

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

      3. unsub-neg 97.3%

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

      4. metadata-eval 97.3%

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

      5. metadata-eval 97.3%

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

      6. associate-*r/ 97.3%

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

      7. metadata-eval 97.3%

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

      8. metadata-eval 97.3%

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

   6. Simplified 97.3%

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

   if -0.5 < x < 1

   1. Initial program 22.1%

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

      1. +-commutative 22.1%

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

      2. hypot-1-def 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in x around 0 22.8%

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

      1. log1p-def 99.5%

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

      2. unpow2 99.5%

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

   6. Simplified 99.5%

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

      1. add-sqr-sqrt_binary32 98.0%

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

   8. Applied rewrite-once 98.0%

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

   10. Applied egg-rr 99.5%

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

   if 1 < x

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-1-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. associate-*r/ 97.6%

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

      2. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

4. Final simplification 98.6%

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

Alternative 6: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 55.8%

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

      1. +-commutative 55.8%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. unsub-neg 96.5%

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

   6. Simplified 96.5%

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

   if -0.5 < x < 1

   1. Initial program 22.1%

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

      1. +-commutative 22.1%

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

      2. hypot-1-def 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in x around 0 22.8%

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

      1. log1p-def 99.5%

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

      2. unpow2 99.5%

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

   6. Simplified 99.5%

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

      1. add-sqr-sqrt_binary32 98.0%

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

   8. Applied rewrite-once 98.0%

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

   10. Applied egg-rr 99.5%

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

   if 1 < x

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-1-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. associate-*r/ 97.6%

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

      2. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in x around 0 96.2%

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

4. Final simplification 98.1%

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

Alternative 7: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 55.8%

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

      1. +-commutative 55.8%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 97.3%

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

      1. cancel-sign-sub-inv 97.3%

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

      2. mul-1-neg 97.3%

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

      3. unsub-neg 97.3%

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

      4. metadata-eval 97.3%

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

      5. metadata-eval 97.3%

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

      6. associate-*r/ 97.3%

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

      7. metadata-eval 97.3%

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

      8. metadata-eval 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in x around 0 96.5%

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

   if -0.5 < x < 1

   1. Initial program 22.1%

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

      1. +-commutative 22.1%

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

      2. hypot-1-def 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in x around 0 22.8%

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

      1. log1p-def 99.5%

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

      2. unpow2 99.5%

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

   6. Simplified 99.5%

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

      1. add-sqr-sqrt_binary32 98.0%

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

   8. Applied rewrite-once 98.0%

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

   10. Applied egg-rr 99.5%

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

   if 1 < x

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-1-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. associate-*r/ 97.6%

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

      2. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in x around 0 96.2%

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

4. Final simplification 98.1%

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

Alternative 8: 97.5% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0) (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 <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 54.6%

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

      1. +-commutative 54.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 99.1%

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

      1. cancel-sign-sub-inv 99.1%

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

      2. mul-1-neg 99.1%

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

      3. unsub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. metadata-eval 99.1%

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

      6. associate-*r/ 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in x around 0 98.5%

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

   if -2 < x < 1

   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. Taylor expanded in x around 0 20.8%

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

      1. log1p-def 93.1%

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

   6. Simplified 93.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]
   7. Step-by-step derivation

      1. log1p-udef 20.8%

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

      2. flip-+ 20.9%

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

      3. log-div 21.2%

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

      4. metadata-eval 21.2%

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

      5. sub-neg 21.2%

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

      6. sqr-abs 21.2%

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

      7. log1p-def 22.2%

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

      8. distribute-rgt-neg-in 22.2%

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

      9. sub-neg 22.2%

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

      10. log1p-def 93.1%

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

   8. Applied egg-rr 93.1%

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

   9. Taylor expanded in x around 0 21.0%

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

       1. sub-neg 21.0%

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

       2. log1p-def 93.0%

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

       3. neg-mul-1 93.0%

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

       4. unpow1 93.0%

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

       5. sqr-pow 31.0%

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

       6. fabs-sqr 31.0%

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

       7. sqr-pow 93.1%

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

       8. unpow1 93.1%

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

   11. Simplified 93.1%

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

   12. Taylor expanded in x around 0 98.0%

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

       1. neg-mul-1 98.0%

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

   14. Simplified 98.0%

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

   if 1 < x

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-1-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. associate-*r/ 97.6%

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

      2. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in x around 0 96.2%

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

4. Final simplification 97.8%

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

Alternative 9: 82.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.5) (copysign (log (/ -0.5 x)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -0.5f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.5))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -0.5

   1. Initial program 55.8%

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

      1. +-commutative 55.8%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 97.3%

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

      1. cancel-sign-sub-inv 97.3%

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

      2. mul-1-neg 97.3%

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

      3. unsub-neg 97.3%

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

      4. metadata-eval 97.3%

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

      5. metadata-eval 97.3%

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

      6. associate-*r/ 97.3%

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

      7. metadata-eval 97.3%

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

      8. metadata-eval 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in x around 0 96.5%

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

   if -0.5 < x

   1. Initial program 30.0%

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

      1. +-commutative 30.0%

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

      2. hypot-1-def 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in x around 0 19.8%

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

      1. log1p-def 76.3%

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

      2. unpow2 76.3%

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

   6. Simplified 76.3%

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

      1. add-sqr-sqrt_binary32 75.2%

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

   8. Applied rewrite-once 75.2%

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

   9. Taylor expanded in x around 0 26.7%

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

       1. log1p-def 80.7%

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

       2. unpow1 80.7%

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

       3. sqr-pow 34.7%

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

       4. fabs-sqr 34.7%

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

       5. sqr-pow 80.7%

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

       6. unpow1 80.7%

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

   11. Simplified 80.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 10: 68.6% 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 54.6%

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

      1. +-commutative 54.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 44.1%

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

      1. mul-1-neg 44.1%

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

   6. Simplified 44.1%

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

   if -2 < x

   1. Initial program 30.7%

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

      1. +-commutative 30.7%

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

      2. hypot-1-def 42.8%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in x around 0 20.0%

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

      1. log1p-def 75.9%

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

      2. unpow2 75.9%

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

   6. Simplified 75.9%

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

      1. add-sqr-sqrt_binary32 74.8%

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

   8. Applied rewrite-once 74.8%

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

   9. Taylor expanded in x around 0 26.8%

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

       1. log1p-def 80.2%

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

       2. unpow1 80.2%

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

       3. sqr-pow 34.3%

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

       4. fabs-sqr 34.3%

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

       5. sqr-pow 80.2%

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

       6. unpow1 80.2%

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

   11. Simplified 80.2%

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

4. Final simplification 70.6%

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

Alternative 11: 62.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 33.5%

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

      1. +-commutative 33.5%

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

      2. hypot-1-def 48.5%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in x around 0 28.5%

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

      1. log1p-def 77.0%

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

   6. Simplified 77.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]
   7. Step-by-step derivation

      1. log1p-udef 28.5%

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

      2. flip-+ 22.1%

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

      3. log-div 14.2%

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

      4. metadata-eval 14.2%

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

      5. sub-neg 14.2%

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

      6. sqr-abs 14.2%

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

      7. log1p-def 14.9%

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

      8. distribute-rgt-neg-in 14.9%

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

      9. sub-neg 14.9%

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

      10. log1p-def 62.5%

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

   8. Applied egg-rr 62.5%

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

   9. Taylor expanded in x around 0 14.1%

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

       1. sub-neg 14.1%

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

       2. log1p-def 62.4%

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

       3. neg-mul-1 62.4%

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

       4. unpow1 62.4%

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

       5. sqr-pow 20.8%

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

       6. fabs-sqr 20.8%

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

       7. sqr-pow 77.0%

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

       8. unpow1 77.0%

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

   11. Simplified 77.0%

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

   12. Taylor expanded in x around 0 69.5%

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

       1. neg-mul-1 69.5%

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

   14. Simplified 69.5%

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

   if 1 < x

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-1-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in x around inf 43.7%

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

      1. mul-1-neg 43.7%

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

      2. log-rec 43.7%

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

      3. remove-double-neg 43.7%

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

   6. Simplified 43.7%

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

4. Final simplification 64.6%

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

Alternative 12: 62.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 33.5%

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

      1. +-commutative 33.5%

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

      2. hypot-1-def 48.5%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in x around 0 28.5%

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

      1. log1p-def 77.0%

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

   6. Simplified 77.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]
   7. Step-by-step derivation

      1. log1p-udef 28.5%

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

      2. flip-+ 22.1%

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

      3. log-div 14.2%

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

      4. metadata-eval 14.2%

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

      5. sub-neg 14.2%

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

      6. sqr-abs 14.2%

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

      7. log1p-def 14.9%

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

      8. distribute-rgt-neg-in 14.9%

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

      9. sub-neg 14.9%

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

      10. log1p-def 62.5%

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

   8. Applied egg-rr 62.5%

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

   9. Taylor expanded in x around 0 14.1%

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

       1. sub-neg 14.1%

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

       2. log1p-def 62.4%

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

       3. neg-mul-1 62.4%

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

       4. unpow1 62.4%

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

       5. sqr-pow 20.8%

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

       6. fabs-sqr 20.8%

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

       7. sqr-pow 77.0%

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

       8. unpow1 77.0%

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

   11. Simplified 77.0%

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

   12. Taylor expanded in x around 0 69.5%

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

       1. neg-mul-1 69.5%

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

   14. Simplified 69.5%

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

   if 1 < x

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-1-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in x around 0 11.5%

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

      1. log1p-def 11.5%

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

      2. unpow2 11.5%

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

   6. Simplified 11.5%

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

      1. add-sqr-sqrt_binary32 11.5%

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

   8. Applied rewrite-once 11.5%

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

   9. Taylor expanded in x around 0 43.7%

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

       1. log1p-def 43.7%

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

       2. unpow1 43.7%

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

       3. sqr-pow 43.7%

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

       4. fabs-sqr 43.7%

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

       5. sqr-pow 43.7%

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

       6. unpow1 43.7%

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

   11. Simplified 43.7%

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

4. Final simplification 64.6%

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

Alternative 13: 8.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(0, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign 0.0 x))

C

float code(float x) {
	return copysignf(0.0f, x);
}

Julia

function code(x)
	return copysign(Float32(0.0), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(0.0));
end

Derivation

1. Initial program 37.1%

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

   1. +-commutative 37.1%

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

   2. hypot-1-def 58.0%

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

3. Simplified 58.0%

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

4. Taylor expanded in x around -inf 30.2%

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

   1. cancel-sign-sub-inv 30.2%

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

   2. mul-1-neg 30.2%

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

   3. unsub-neg 30.2%

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

   4. metadata-eval 30.2%

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

   5. metadata-eval 30.2%

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

   6. associate-*r/ 30.2%

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

   7. metadata-eval 30.2%

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

   8. metadata-eval 30.2%

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

6. Simplified 30.2%

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

8. Applied egg-rr -0.0%

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

   1. +-inverses 8.6%

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

10. Simplified 8.6%

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

11. Final simplification 8.6%

    \[\leadsto \mathsf{copysign}\left(0, x\right) \]

Alternative 14: 53.9% 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 37.1%

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

   1. +-commutative 37.1%

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

   2. hypot-1-def 58.0%

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

3. Simplified 58.0%

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

4. Taylor expanded in x around 0 31.4%

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

   1. log1p-def 70.7%

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

6. Simplified 70.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]
7. Step-by-step derivation

   1. log1p-udef 31.4%

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

   2. flip-+ 22.3%

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

   3. log-div 11.5%

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

   4. metadata-eval 11.5%

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

   5. sub-neg 11.5%

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

   6. sqr-abs 11.5%

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

   7. log1p-def 12.0%

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

   8. distribute-rgt-neg-in 12.0%

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

   9. sub-neg 12.0%

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

   10. log1p-def 50.5%

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

8. Applied egg-rr 50.5%

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

9. Taylor expanded in x around 0 11.4%

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

    1. sub-neg 11.4%

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

    2. log1p-def 50.5%

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

    3. neg-mul-1 50.5%

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

    4. unpow1 50.5%

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

    5. sqr-pow 16.8%

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

    6. fabs-sqr 16.8%

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

    7. sqr-pow 62.3%

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

    8. unpow1 62.3%

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

11. Simplified 62.3%

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

12. Taylor expanded in x around 0 58.5%

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

    1. neg-mul-1 58.5%

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

14. Simplified 58.5%

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

15. Final simplification 58.5%

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

Developer target: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

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