Rust f32::asinh

Percentage Accurate: 38.3% → 98.5%

Time: 8.9s

Alternatives: 8

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: 38.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.03999999910593033)
     (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 <= -100.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.03999999910593033f) {
		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(-100.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.03999999910593033))
		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(-100.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.03999999910593033))
		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 < -100

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. mul-1-neg 98.7%

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

      4. unsub-neg 98.7%

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

      5. sub-neg 98.7%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 8.8%

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

      9. associate-*r/ 8.8%

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

      10. metadata-eval 8.8%

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

      11. distribute-neg-frac 8.8%

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

      12. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -100 < x < 0.0399999991

   1. Initial program 25.0%

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

   3. Taylor expanded in x around 0 25.3%

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

      1. +-commutative 25.3%

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

      2. fma-define 25.3%

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

      3. rem-square-sqrt 10.5%

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

      4. fabs-sqr 10.5%

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

      5. rem-square-sqrt 24.7%

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

      6. log1p-define 97.8%

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

      7. rem-square-sqrt 44.3%

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

      8. fabs-sqr 44.3%

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

      9. rem-square-sqrt 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. unpow2 99.5%

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

      6. cube-mult 99.5%

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

   8. Simplified 99.5%

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

   if 0.0399999991 < x

   1. Initial program 53.8%

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

      1. +-commutative 53.8%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around 0 53.8%

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

      1. rem-square-sqrt 53.8%

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

      2. fabs-sqr 53.8%

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

      3. rem-square-sqrt 53.8%

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

      4. metadata-eval 53.8%

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

      5. unpow2 53.8%

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

      6. hypot-undefine 98.5%

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

   7. Simplified 98.5%

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

Alternative 2: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(3 \cdot \left(0.3333333333333333 \cdot \mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -5.0)
   (copysign (log (/ -0.5 x)) x)
   (copysign
    (* 3.0 (* 0.3333333333333333 (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) <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf((3.0f * (0.3333333333333333f * 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(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(Float32(Float32(3.0) * Float32(Float32(0.3333333333333333) * 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) #s(literal 1 binary32))))) x) < -5

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. mul-1-neg 98.7%

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

      4. unsub-neg 98.7%

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

      5. sub-neg 98.7%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 8.8%

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

      9. associate-*r/ 8.8%

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

      10. metadata-eval 8.8%

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

      11. distribute-neg-frac 8.8%

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

      12. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 35.4%

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

      1. add-cube-cbrt 34.3%

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

      2. pow3 34.3%

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

      3. log-pow 34.1%

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

      4. *-un-lft-identity 34.1%

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

      5. *-un-lft-identity 34.1%

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

      6. add-sqr-sqrt 25.3%

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

      7. fabs-sqr 25.3%

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

      8. add-sqr-sqrt 34.4%

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

      9. +-commutative 34.4%

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

      10. hypot-1-def 50.5%

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

   4. Applied egg-rr 50.5%

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

      1. pow1/3 50.4%

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

      2. log-pow 51.3%

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

   6. Applied egg-rr 51.3%

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

      1. log1p-expm1-u 51.2%

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

      2. *-un-lft-identity 51.2%

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

      3. metadata-eval 51.2%

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

      4. associate-*r* 51.2%

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

      5. expm1-undefine 51.2%

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

      6. associate-*r* 51.2%

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

      7. metadata-eval 51.2%

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

      8. *-un-lft-identity 51.2%

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

      9. add-exp-log 51.2%

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

   8. Applied egg-rr 51.2%

      \[\leadsto \mathsf{copysign}\left(3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)}\right), x\right) \]
   9. Step-by-step derivation

      1. associate--l+ 96.2%

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

   10. Simplified 96.2%

       \[\leadsto \mathsf{copysign}\left(3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)}\right), x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(3 \cdot \left(0.3333333333333333 \cdot \mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (- (log (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -100.0f) {
		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((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-100.0))
		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(Float32(-log(Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-100.0))
		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((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -100

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. mul-1-neg 98.7%

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

      4. unsub-neg 98.7%

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

      5. sub-neg 98.7%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 8.8%

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

      9. associate-*r/ 8.8%

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

      10. metadata-eval 8.8%

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

      11. distribute-neg-frac 8.8%

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

      12. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -100 < x < 1

   1. Initial program 26.8%

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

   3. Taylor expanded in x around 0 26.1%

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

      1. +-commutative 26.1%

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

      2. fma-define 26.1%

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

      3. rem-square-sqrt 11.6%

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

      4. fabs-sqr 11.6%

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

      5. rem-square-sqrt 25.5%

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

      6. log1p-define 96.8%

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

      7. rem-square-sqrt 44.6%

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

      8. fabs-sqr 44.6%

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

      9. rem-square-sqrt 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. distribute-lft-in 98.7%

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

      2. *-rgt-identity 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-*r* 98.7%

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

      5. unpow2 98.7%

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

      6. cube-mult 98.7%

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

   8. Simplified 98.7%

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

   if 1 < x

   1. Initial program 51.8%

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

      1. flip-+ 3.1%

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

      2. clear-num 3.1%

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

      3. log-div 3.1%

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

      4. metadata-eval 3.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. +-commutative 3.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. hypot-1-def 3.1%

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

      7. add-sqr-sqrt 3.5%

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

      8. fabs-sqr 3.5%

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

      9. add-sqr-sqrt 3.1%

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

   4. Applied egg-rr 2.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 2.9%

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

      2. div-sub 2.9%

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

      3. *-rgt-identity 2.9%

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

      4. associate-/l* 2.9%

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

      5. fma-neg 2.9%

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

      6. fma-undefine 2.9%

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

      7. unpow2 2.9%

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

      8. associate--r+ 2.9%

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

      9. +-inverses 2.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\mathsf{fma}\left(x, \frac{1}{\color{blue}{0} - 1}, -\frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)\right), x\right) \]

      10. metadata-eval 2.9%

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

      11. metadata-eval 2.9%

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

      12. *-rgt-identity 2.9%

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

      13. associate-/l* 2.9%

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

      14. fma-undefine 2.9%

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

      15. unpow2 2.9%

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

      16. associate--r+ 5.6%

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

   6. Simplified 8.9%

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

   7. Taylor expanded in x around inf 99.9%

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

Alternative 4: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (- (log (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -100.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(-logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-100.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-100.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(-log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -100

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. mul-1-neg 98.7%

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

      4. unsub-neg 98.7%

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

      5. sub-neg 98.7%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 8.8%

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

      9. associate-*r/ 8.8%

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

      10. metadata-eval 8.8%

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

      11. distribute-neg-frac 8.8%

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

      12. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -100 < x < 1

   1. Initial program 26.8%

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

      1. flip-+ 26.6%

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

      2. clear-num 26.6%

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

      3. log-div 26.9%

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

      4. metadata-eval 26.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. +-commutative 26.9%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. hypot-1-def 26.7%

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

      7. add-sqr-sqrt 12.0%

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

      8. fabs-sqr 12.0%

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

      9. add-sqr-sqrt 26.5%

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

   4. Applied egg-rr 26.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 26.5%

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

      2. div-sub 26.5%

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

      3. *-rgt-identity 26.5%

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

      4. associate-/l* 26.5%

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

      5. fma-neg 26.5%

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

      6. fma-undefine 26.5%

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

      7. unpow2 26.5%

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

      8. associate--r+ 26.5%

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

      9. +-inverses 26.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\mathsf{fma}\left(x, \frac{1}{\color{blue}{0} - 1}, -\frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)\right), x\right) \]

      10. metadata-eval 26.5%

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

      11. metadata-eval 26.5%

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

      12. *-rgt-identity 26.5%

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

      13. associate-/l* 26.5%

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

      14. fma-undefine 26.5%

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

      15. unpow2 26.5%

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

      16. associate--r+ 26.8%

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

   6. Simplified 26.8%

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

   7. Taylor expanded in x around 0 96.4%

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

   if 1 < x

   1. Initial program 51.8%

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

      1. flip-+ 3.1%

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

      2. clear-num 3.1%

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

      3. log-div 3.1%

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

      4. metadata-eval 3.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. +-commutative 3.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. hypot-1-def 3.1%

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

      7. add-sqr-sqrt 3.5%

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

      8. fabs-sqr 3.5%

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

      9. add-sqr-sqrt 3.1%

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

   4. Applied egg-rr 2.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 2.9%

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

      2. div-sub 2.9%

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

      3. *-rgt-identity 2.9%

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

      4. associate-/l* 2.9%

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

      5. fma-neg 2.9%

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

      6. fma-undefine 2.9%

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

      7. unpow2 2.9%

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

      8. associate--r+ 2.9%

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

      9. +-inverses 2.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\mathsf{fma}\left(x, \frac{1}{\color{blue}{0} - 1}, -\frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)\right), x\right) \]

      10. metadata-eval 2.9%

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

      11. metadata-eval 2.9%

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

      12. *-rgt-identity 2.9%

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

      13. associate-/l* 2.9%

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

      14. fma-undefine 2.9%

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

      15. unpow2 2.9%

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

      16. associate--r+ 5.6%

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

   6. Simplified 8.9%

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

   7. Taylor expanded in x around inf 99.9%

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

Alternative 5: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0)
   (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 <= -100.0f) {
		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(-100.0))
		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(-100.0))
		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 < -100

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

      3. mul-1-neg 98.7%

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

      4. unsub-neg 98.7%

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

      5. sub-neg 98.7%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 8.8%

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

      9. associate-*r/ 8.8%

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

      10. metadata-eval 8.8%

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

      11. distribute-neg-frac 8.8%

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

      12. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -100 < x < 1

   1. Initial program 26.8%

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

      1. flip-+ 26.6%

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

      2. clear-num 26.6%

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

      3. log-div 26.9%

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

      4. metadata-eval 26.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. +-commutative 26.9%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. hypot-1-def 26.7%

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

      7. add-sqr-sqrt 12.0%

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

      8. fabs-sqr 12.0%

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

      9. add-sqr-sqrt 26.5%

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

   4. Applied egg-rr 26.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 26.5%

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

      2. div-sub 26.5%

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

      3. *-rgt-identity 26.5%

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

      4. associate-/l* 26.5%

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

      5. fma-neg 26.5%

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

      6. fma-undefine 26.5%

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

      7. unpow2 26.5%

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

      8. associate--r+ 26.5%

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

      9. +-inverses 26.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\mathsf{fma}\left(x, \frac{1}{\color{blue}{0} - 1}, -\frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)\right), x\right) \]

      10. metadata-eval 26.5%

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

      11. metadata-eval 26.5%

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

      12. *-rgt-identity 26.5%

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

      13. associate-/l* 26.5%

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

      14. fma-undefine 26.5%

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

      15. unpow2 26.5%

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

      16. associate--r+ 26.8%

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

   6. Simplified 26.8%

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

   7. Taylor expanded in x around 0 96.4%

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

   if 1 < x

   1. Initial program 51.8%

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

   3. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. rem-square-sqrt 98.5%

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

      3. fabs-sqr 98.5%

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

      4. rem-square-sqrt 98.5%

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

      5. *-inverses 98.5%

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

      6. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

Alternative 6: 75.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	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(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(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 2 regimes

2. if x < 1

   1. Initial program 36.5%

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

      1. flip-+ 17.8%

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

      2. clear-num 17.8%

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

      3. log-div 18.1%

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

      4. metadata-eval 18.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. +-commutative 18.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. hypot-1-def 17.9%

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

      7. add-sqr-sqrt 7.6%

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

      8. fabs-sqr 7.6%

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

      9. add-sqr-sqrt 17.7%

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

   4. Applied egg-rr 18.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 18.4%

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

      2. div-sub 18.4%

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

      3. *-rgt-identity 18.4%

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

      4. associate-/l* 18.4%

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

      5. fma-neg 18.4%

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

      6. fma-undefine 18.4%

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

      7. unpow2 18.4%

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

      8. associate--r+ 18.4%

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

      9. +-inverses 18.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\mathsf{fma}\left(x, \frac{1}{\color{blue}{0} - 1}, -\frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)\right), x\right) \]

      10. metadata-eval 18.4%

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

      11. metadata-eval 18.4%

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

      12. *-rgt-identity 18.4%

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

      13. associate-/l* 18.4%

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

      14. fma-undefine 18.4%

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

      15. unpow2 18.4%

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

      16. associate--r+ 35.3%

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

   6. Simplified 53.0%

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

   7. Taylor expanded in x around 0 65.1%

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

   if 1 < x

   1. Initial program 51.8%

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

   3. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. rem-square-sqrt 98.5%

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

      3. fabs-sqr 98.5%

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

      4. rem-square-sqrt 98.5%

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

      5. *-inverses 98.5%

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

      6. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

Alternative 7: 61.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 36.5%

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

      1. flip-+ 17.8%

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

      2. clear-num 17.8%

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

      3. log-div 18.1%

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

      4. metadata-eval 18.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. +-commutative 18.1%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. hypot-1-def 17.9%

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

      7. add-sqr-sqrt 7.6%

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

      8. fabs-sqr 7.6%

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

      9. add-sqr-sqrt 17.7%

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

   4. Applied egg-rr 18.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 18.4%

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

      2. div-sub 18.4%

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

      3. *-rgt-identity 18.4%

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

      4. associate-/l* 18.4%

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

      5. fma-neg 18.4%

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

      6. fma-undefine 18.4%

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

      7. unpow2 18.4%

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

      8. associate--r+ 18.4%

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

      9. +-inverses 18.4%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\mathsf{fma}\left(x, \frac{1}{\color{blue}{0} - 1}, -\frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)\right), x\right) \]

      10. metadata-eval 18.4%

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

      11. metadata-eval 18.4%

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

      12. *-rgt-identity 18.4%

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

      13. associate-/l* 18.4%

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

      14. fma-undefine 18.4%

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

      15. unpow2 18.4%

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

      16. associate--r+ 35.3%

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

   6. Simplified 53.0%

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

   7. Taylor expanded in x around 0 65.1%

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

   if 1 < x

   1. Initial program 51.8%

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

   3. Taylor expanded in x around 0 44.5%

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

      1. log1p-define 44.5%

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

      2. rem-square-sqrt 44.5%

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

      3. fabs-sqr 44.5%

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

      4. rem-square-sqrt 44.5%

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

   5. Simplified 44.5%

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

Alternative 8: 53.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 40.3%

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

   1. flip-+ 14.2%

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

   2. clear-num 14.2%

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

   3. log-div 14.3%

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

   4. metadata-eval 14.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0} - \log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

   5. +-commutative 14.3%

      \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

   6. hypot-1-def 14.2%

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

   7. add-sqr-sqrt 6.6%

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

   8. fabs-sqr 6.6%

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

   9. add-sqr-sqrt 14.1%

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

4. Applied egg-rr 14.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
5. Step-by-step derivation

   1. neg-sub0 14.5%

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

   2. div-sub 14.5%

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

   3. *-rgt-identity 14.5%

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

   4. associate-/l* 14.5%

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

   5. fma-neg 14.5%

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

   6. fma-undefine 14.5%

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

   7. unpow2 14.5%

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

   8. associate--r+ 14.5%

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

   9. +-inverses 14.5%

      \[\leadsto \mathsf{copysign}\left(-\log \left(\mathsf{fma}\left(x, \frac{1}{\color{blue}{0} - 1}, -\frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)\right), x\right) \]

   10. metadata-eval 14.5%

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

   11. metadata-eval 14.5%

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

   12. *-rgt-identity 14.5%

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

   13. associate-/l* 14.5%

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

   14. fma-undefine 14.5%

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

   15. unpow2 14.5%

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

   16. associate--r+ 27.9%

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

6. Simplified 42.0%

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

7. Taylor expanded in x around 0 51.5%

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

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024137 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

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

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