Rust f32::asinh

Percentage Accurate: 37.6% → 98.7%

Time: 10.9s

Alternatives: 13

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 48.3%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. sub-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 -0.0%

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

      1. neg-mul-1 -0.0%

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

      2. unsub-neg -0.0%

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

      3. log-div 99.0%

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

   8. Simplified 99.0%

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

   if -10 < x < 0.0500000007

   1. Initial program 19.3%

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

   4. Applied egg-rr 19.6%

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

      1. +-commutative 19.6%

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

      2. fma-udef 19.6%

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

      3. unpow2 19.6%

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

      4. +-commutative 19.6%

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

      5. associate-+l+ 19.6%

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

      6. log1p-def 97.3%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      7. unpow2 97.3%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      8. distribute-lft-out-- 97.3%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. Simplified 97.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

   9. Simplified 99.3%

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

   if 0.0500000007 < x

   1. Initial program 61.9%

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

      1. *-un-lft-identity 61.9%

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

      2. *-commutative 61.9%

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

      3. log-prod 61.9%

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

      4. +-commutative 61.9%

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

      5. hypot-1-def 98.1%

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

      6. add-sqr-sqrt 98.0%

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

      7. fabs-sqr 98.0%

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

      8. add-sqr-sqrt 98.1%

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

      9. metadata-eval 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. +-rgt-identity 98.1%

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

   6. Simplified 98.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -2.0)
   (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x) <= Float32(-2.0))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(hypot(Float32(1.0), x) + Float32(-1.0)))), 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) < -2

   1. Initial program 48.3%

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

      1. +-commutative 48.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

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

   1. Initial program 33.2%

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

      1. *-un-lft-identity 33.2%

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

      2. add-log-exp 18.4%

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

      3. exp-prod 18.2%

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

      4. log-pow 32.9%

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

      5. +-commutative 32.9%

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

      6. hypot-1-def 44.8%

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

      7. add-sqr-sqrt 38.7%

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

      8. fabs-sqr 38.7%

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

      9. add-sqr-sqrt 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. log1p-expm1-u 44.9%

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

      2. expm1-udef 44.9%

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

      3. rem-log-exp 45.3%

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

      4. add-exp-log 45.3%

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

      5. *-rgt-identity 45.3%

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

   6. Applied egg-rr 45.3%

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

      1. associate--l+ 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 98.5%

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

Alternative 3: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else {
		tmp = copysignf(log1pf((x + (hypotf(1.0f, x) + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(hypot(Float32(1.0), x) + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -10

   1. Initial program 48.3%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. sub-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   if -10 < x

   1. Initial program 33.2%

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

      1. *-un-lft-identity 33.2%

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

      2. add-log-exp 18.4%

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

      3. exp-prod 18.2%

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

      4. log-pow 32.9%

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

      5. +-commutative 32.9%

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

      6. hypot-1-def 44.8%

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

      7. add-sqr-sqrt 38.7%

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

      8. fabs-sqr 38.7%

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

      9. add-sqr-sqrt 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. log1p-expm1-u 44.9%

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

      2. expm1-udef 44.9%

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

      3. rem-log-exp 45.3%

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

      4. add-exp-log 45.3%

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

      5. *-rgt-identity 45.3%

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

   6. Applied egg-rr 45.3%

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

      1. associate--l+ 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (- (fabs x) (+ x (/ 0.5 x)))) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((fabsf(x) - (x + (0.5f / x)))), x);
	} else {
		tmp = copysignf(log1pf((x + (hypotf(1.0f, x) + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(abs(x) - Float32(x + Float32(Float32(0.5) / x)))), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(hypot(Float32(1.0), x) + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -10

   1. Initial program 48.3%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. neg-mul-1 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   if -10 < x

   1. Initial program 33.2%

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

      1. *-un-lft-identity 33.2%

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

      2. add-log-exp 18.4%

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

      3. exp-prod 18.2%

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

      4. log-pow 32.9%

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

      5. +-commutative 32.9%

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

      6. hypot-1-def 44.8%

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

      7. add-sqr-sqrt 38.7%

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

      8. fabs-sqr 38.7%

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

      9. add-sqr-sqrt 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. log1p-expm1-u 44.9%

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

      2. expm1-udef 44.9%

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

      3. rem-log-exp 45.3%

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

      4. add-exp-log 45.3%

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

      5. *-rgt-identity 45.3%

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

   6. Applied egg-rr 45.3%

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

      1. associate--l+ 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -400:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -400.0)
   (copysign (log (/ -0.5 x)) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

float code(float x) {
	float tmp;
	if (x <= -400.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf(log1pf((x + (hypotf(1.0f, x) + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-400.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(hypot(Float32(1.0), x) + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -400

   1. Initial program 46.8%

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

   3. Taylor expanded in x around -inf 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 -0.0%

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

      1. neg-mul-1 -0.0%

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

      2. unsub-neg -0.0%

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

      3. log-div 99.9%

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

   8. Simplified 99.9%

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

   if -400 < x

   1. Initial program 33.9%

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

      1. *-un-lft-identity 33.9%

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

      2. add-log-exp 18.8%

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

      3. exp-prod 18.6%

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

      4. log-pow 33.6%

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

      5. +-commutative 33.6%

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

      6. hypot-1-def 45.4%

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

      7. add-sqr-sqrt 38.3%

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

      8. fabs-sqr 38.3%

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

      9. add-sqr-sqrt 45.3%

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

   4. Applied egg-rr 45.3%

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

      1. log1p-expm1-u 45.3%

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

      2. expm1-udef 45.3%

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

      3. rem-log-exp 45.6%

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

      4. add-exp-log 45.7%

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

      5. *-rgt-identity 45.7%

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

   6. Applied egg-rr 45.7%

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

      1. associate--l+ 97.8%

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

   8. Simplified 97.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -400:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x + \left(\frac{0.5}{x} + -1\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log1p (+ x (+ x (+ (/ 0.5 x) -1.0)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(log1pf((x + (x + ((0.5f / x) + -1.0f)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(x + Float32(Float32(Float32(0.5) / x) + Float32(-1.0))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 48.3%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. sub-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 -0.0%

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

      1. neg-mul-1 -0.0%

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

      2. unsub-neg -0.0%

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

      3. log-div 99.0%

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

   8. Simplified 99.0%

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

   if -10 < x < 1

   1. Initial program 21.6%

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

   4. Applied egg-rr 21.8%

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

      1. +-commutative 21.8%

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

      2. fma-udef 21.7%

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

      3. unpow2 21.7%

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

      4. +-commutative 21.7%

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

      5. associate-+l+ 21.8%

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

      6. log1p-def 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      7. unpow2 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      8. distribute-lft-out-- 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. Simplified 97.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   9. Simplified 98.3%

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

   if 1 < x

   1. Initial program 59.6%

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

      1. *-un-lft-identity 59.6%

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

      2. add-log-exp 9.7%

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

      3. exp-prod 9.7%

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

      4. log-pow 59.6%

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

      5. +-commutative 59.6%

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

      6. hypot-1-def 98.3%

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

      7. add-sqr-sqrt 98.3%

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

      8. fabs-sqr 98.3%

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

      9. add-sqr-sqrt 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. log1p-expm1-u 98.3%

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

      2. expm1-udef 98.3%

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

      3. rem-log-exp 98.3%

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

      4. add-exp-log 98.3%

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

      5. *-rgt-identity 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. associate--l+ 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in x around inf 97.1%

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

       1. associate--l+ 97.1%

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

       2. associate-*r/ 97.1%

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

       3. metadata-eval 97.1%

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

   11. Simplified 97.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x + \left(\frac{0.5}{x} + -1\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 48.3%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. sub-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 -0.0%

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

      1. neg-mul-1 -0.0%

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

      2. unsub-neg -0.0%

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

      3. log-div 99.0%

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

   8. Simplified 99.0%

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

   if -10 < x < 1

   1. Initial program 21.6%

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

   4. Applied egg-rr 21.8%

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

      1. +-commutative 21.8%

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

      2. fma-udef 21.7%

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

      3. unpow2 21.7%

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

      4. +-commutative 21.7%

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

      5. associate-+l+ 21.8%

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

      6. log1p-def 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      7. unpow2 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      8. distribute-lft-out-- 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. Simplified 97.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   9. Simplified 98.3%

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

   if 1 < x

   1. Initial program 59.6%

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

   4. Applied egg-rr 36.5%

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

      1. +-commutative 36.5%

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

      2. fma-udef 36.5%

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

      3. unpow2 36.5%

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

      4. +-commutative 36.5%

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

      5. associate-+l+ 36.5%

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

      6. log1p-def 36.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      7. unpow2 36.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      8. distribute-lft-out-- 36.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. Simplified 36.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around inf 94.7%

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

      1. associate--l+ 95.0%

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

      2. distribute-rgt-out-- 96.4%

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

      3. metadata-eval 96.4%

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

      4. *-commutative 96.4%

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

      5. +-commutative 96.4%

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

      6. mul-1-neg 96.4%

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

      7. log-rec 96.5%

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

      8. remove-double-neg 96.5%

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

      9. log-prod 95.5%

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

   9. Simplified 95.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (- x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 48.3%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. sub-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf -0.0%

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

      1. mul-1-neg -0.0%

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

      2. log-rec -0.0%

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

      3. remove-double-neg -0.0%

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

      4. log-prod 44.7%

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

      5. neg-mul-1 44.7%

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

   8. Simplified 44.7%

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

   if -10 < x < 1

   1. Initial program 21.6%

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

   4. Applied egg-rr 21.8%

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

      1. +-commutative 21.8%

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

      2. fma-udef 21.7%

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

      3. unpow2 21.7%

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

      4. +-commutative 21.7%

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

      5. associate-+l+ 21.8%

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

      6. log1p-def 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      7. unpow2 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      8. distribute-lft-out-- 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. Simplified 97.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 96.6%

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

   if 1 < x

   1. Initial program 59.6%

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

   4. Applied egg-rr 36.5%

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

      1. +-commutative 36.5%

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

      2. fma-udef 36.5%

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

      3. unpow2 36.5%

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

      4. +-commutative 36.5%

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

      5. associate-+l+ 36.5%

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

      6. log1p-def 36.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      7. unpow2 36.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      8. distribute-lft-out-- 36.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. Simplified 36.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around inf 94.7%

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

      1. associate--l+ 95.0%

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

      2. distribute-rgt-out-- 96.4%

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

      3. metadata-eval 96.4%

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

      4. *-commutative 96.4%

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

      5. +-commutative 96.4%

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

      6. mul-1-neg 96.4%

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

      7. log-rec 96.5%

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

      8. remove-double-neg 96.5%

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

      9. log-prod 95.5%

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

   9. Simplified 95.5%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		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((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 48.3%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. sub-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 -0.0%

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

      1. neg-mul-1 -0.0%

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

      2. unsub-neg -0.0%

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

      3. log-div 99.0%

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

   8. Simplified 99.0%

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

   if -10 < x < 1

   1. Initial program 21.6%

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

   4. Applied egg-rr 21.8%

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

      1. +-commutative 21.8%

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

      2. fma-udef 21.7%

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

      3. unpow2 21.7%

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

      4. +-commutative 21.7%

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

      5. associate-+l+ 21.8%

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

      6. log1p-def 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      7. unpow2 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      8. distribute-lft-out-- 97.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. Simplified 97.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 96.6%

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

   if 1 < x

   1. Initial program 59.6%

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

   4. Applied egg-rr 36.5%

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

      1. +-commutative 36.5%

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

      2. fma-udef 36.5%

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

      3. unpow2 36.5%

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

      4. +-commutative 36.5%

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

      5. associate-+l+ 36.5%

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

      6. log1p-def 36.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      7. unpow2 36.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      8. distribute-lft-out-- 36.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. Simplified 36.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around inf 94.7%

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

      1. associate--l+ 95.0%

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

      2. distribute-rgt-out-- 96.4%

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

      3. metadata-eval 96.4%

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

      4. *-commutative 96.4%

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

      5. +-commutative 96.4%

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

      6. mul-1-neg 96.4%

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

      7. log-rec 96.5%

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

      8. remove-double-neg 96.5%

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

      9. log-prod 95.5%

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

   9. Simplified 95.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0) (copysign (log (- x)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		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 < -10

   1. Initial program 48.3%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. sub-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf -0.0%

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

      1. mul-1-neg -0.0%

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

      2. log-rec -0.0%

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

      3. remove-double-neg -0.0%

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

      4. log-prod 44.7%

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

      5. neg-mul-1 44.7%

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

   8. Simplified 44.7%

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

   if -10 < x

   1. Initial program 33.2%

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

   3. Taylor expanded in x around 0 25.6%

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

      1. log1p-def 78.4%

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

   5. Simplified 78.4%

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

      1. expm1-log1p-u 78.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\left|x\right|\right)\right)\right)}, x\right) \]

      2. expm1-udef 25.6%

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

      3. add-sqr-sqrt 20.5%

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

      4. fabs-sqr 20.5%

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

      5. add-sqr-sqrt 25.7%

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

   7. Applied egg-rr 25.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)} - 1}, x\right) \]
   8. Step-by-step derivation

      1. expm1-def 78.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)\right)}, x\right) \]

      2. expm1-log1p 78.4%

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

   9. Simplified 78.4%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.200000047683716:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 3.200000047683716) (copysign x x) (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 3.200000047683716f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(3.200000047683716))
		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(3.200000047683716))
		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 < 3.20000005

   1. Initial program 31.0%

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

   4. Applied egg-rr 16.8%

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

      1. +-commutative 16.8%

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

      2. fma-udef 16.8%

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

      3. unpow2 16.8%

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

      4. +-commutative 16.8%

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

      5. associate-+l+ 16.8%

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

      6. log1p-def 66.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      7. unpow2 66.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      8. distribute-lft-out-- 66.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. Simplified 66.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 67.3%

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

   if 3.20000005 < x

   1. Initial program 58.9%

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

   3. Taylor expanded in x around inf 44.0%

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

      1. mul-1-neg 44.0%

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

      2. log-rec 44.0%

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

      3. remove-double-neg 44.0%

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

   5. Simplified 44.0%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.200000047683716:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.8% 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 30.6%

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

   4. Applied egg-rr 16.4%

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

      1. +-commutative 16.4%

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

      2. fma-udef 16.3%

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

      3. unpow2 16.3%

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

      4. +-commutative 16.3%

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

      5. associate-+l+ 16.4%

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

      6. log1p-def 66.3%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

      7. unpow2 66.3%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      8. distribute-lft-out-- 66.3%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. Simplified 66.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 67.5%

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

   if 1 < x

   1. Initial program 59.6%

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

   3. Taylor expanded in x around 0 43.9%

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

      1. log1p-def 43.9%

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

   5. Simplified 43.9%

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

      1. expm1-log1p-u 43.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\left|x\right|\right)\right)\right)}, x\right) \]

      2. expm1-udef 43.9%

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

      3. add-sqr-sqrt 43.9%

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

      4. fabs-sqr 43.9%

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

      5. add-sqr-sqrt 43.9%

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

   7. Applied egg-rr 43.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)} - 1}, x\right) \]
   8. Step-by-step derivation

      1. expm1-def 43.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)\right)}, x\right) \]

      2. expm1-log1p 43.9%

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

   9. Simplified 43.9%

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

4. Final simplification 62.1%

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

Alternative 13: 54.7% 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 37.2%

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

4. Applied egg-rr 20.9%

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

   1. +-commutative 20.9%

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

   2. fma-udef 20.9%

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

   3. unpow2 20.9%

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

   4. +-commutative 20.9%

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

   5. associate-+l+ 20.9%

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

   6. log1p-def 59.5%

      \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({x}^{2} + \left({x}^{2} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. unpow2 59.5%

      \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \left(\color{blue}{x \cdot x} - x \cdot \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   8. distribute-lft-out-- 59.5%

      \[\leadsto \mathsf{copysign}\left(\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + \color{blue}{x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

6. Simplified 59.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left({x}^{3} + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}\right) - \mathsf{log1p}\left({x}^{2} + x \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

7. Taylor expanded in x around 0 54.9%

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

8. Final simplification 54.9%

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

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 2024013 
(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))