Rust f32::asinh

Percentage Accurate: 37.1% → 98.0%

Time: 11.4s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-0.125 + \frac{-0.125}{t\_1}}{t\_1}, \frac{0.5}{t\_1}\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (+ (fabs x) 1.0)))
   (if (<= t_0 -2.0)
     (copysign (log (+ (fabs x) (- (/ -0.5 x) x))) x)
     (if (<= t_0 0.03999999910593033)
       (copysign
        (fma
         x
         (* x (fma (* x x) (/ (+ -0.125 (/ -0.125 t_1)) t_1) (/ 0.5 t_1)))
         (log1p (fabs x)))
        x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = fabsf(x) + 1.0f;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf((fabsf(x) + ((-0.5f / x) - x))), x);
	} else if (t_0 <= 0.03999999910593033f) {
		tmp = copysignf(fmaf(x, (x * fmaf((x * x), ((-0.125f + (-0.125f / t_1)) / t_1), (0.5f / t_1))), log1pf(fabsf(x))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = Float32(abs(x) + Float32(1.0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(abs(x) + Float32(Float32(Float32(-0.5) / x) - x))), x);
	elseif (t_0 <= Float32(0.03999999910593033))
		tmp = copysign(fma(x, Float32(x * fma(Float32(x * x), Float32(Float32(Float32(-0.125) + Float32(Float32(-0.125) / t_1)) / t_1), Float32(Float32(0.5) / t_1))), log1p(abs(x))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -2

   1. Initial program 50.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-neg-in N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

      13. lft-mult-inverse N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower--.f32 N/A

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

   5. Applied rewrites 98.0%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0399999991

   1. Initial program 22.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right) + \log \left(1 + \left|x\right|\right)}, x\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right) + \log \left(1 + \left|x\right|\right), x\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)} + \log \left(1 + \left|x\right|\right), x\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right), \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{24} \cdot \frac{{x}^{2} \cdot \left(3 + 3 \cdot \frac{1}{1 + \left|x\right|}\right)}{1 + \left|x\right|} + \color{blue}{\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}}\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-0.125 + \frac{-0.125}{1 + \left|x\right|}}{1 + \left|x\right|}}, \frac{0.5}{1 + \left|x\right|}\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

   if 0.0399999991 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 59.3%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-+.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-0.125 + \frac{-0.125}{\left|x\right| + 1}}{\left|x\right| + 1}, \frac{0.5}{\left|x\right| + 1}\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x \cdot x, \frac{0.5}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (+ (fabs x) (- (/ -0.5 x) x))) x)
     (if (<= t_0 0.03999999910593033)
       (copysign (fma (* x x) (/ 0.5 (+ (fabs x) 1.0)) (log1p (fabs x))) x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf((fabsf(x) + ((-0.5f / x) - x))), x);
	} else if (t_0 <= 0.03999999910593033f) {
		tmp = copysignf(fmaf((x * x), (0.5f / (fabsf(x) + 1.0f)), log1pf(fabsf(x))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(abs(x) + Float32(Float32(Float32(-0.5) / x) - x))), x);
	elseif (t_0 <= Float32(0.03999999910593033))
		tmp = copysign(fma(Float32(x * x), Float32(Float32(0.5) / Float32(abs(x) + Float32(1.0))), log1p(abs(x))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -2

   1. Initial program 50.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-neg-in N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

      13. lft-mult-inverse N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower--.f32 N/A

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

   5. Applied rewrites 98.0%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0399999991

   1. Initial program 22.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-fabs.f32 N/A

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

      14. lower-log1p.f32 N/A

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

      15. lower-fabs.f32 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.0399999991 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 59.3%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-+.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 98.8%

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

Alternative 3: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (+ (fabs x) (- (/ -0.5 x) x))) x)
     (if (<= t_0 0.03999999910593033)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf((fabsf(x) + ((-0.5f / x) - x))), x);
	} else if (t_0 <= 0.03999999910593033f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(abs(x) + Float32(Float32(Float32(-0.5) / x) - x))), x);
	elseif (t_0 <= Float32(0.03999999910593033))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -2

   1. Initial program 50.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. distribute-lft-neg-in N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

      13. lft-mult-inverse N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower--.f32 N/A

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

   5. Applied rewrites 98.0%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0399999991

   1. Initial program 22.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f32 N/A

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

      2. lower-fabs.f32 93.6

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

   5. Applied rewrites 93.6%

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

   if 0.0399999991 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 59.3%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-+.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 4: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.03999999910593033)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (t_0 <= 0.03999999910593033f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.03999999910593033))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -2

   1. Initial program 50.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. lower--.f32 N/A

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

      16. lower-fabs.f32 96.3

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

   5. Applied rewrites 96.3%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0399999991

   1. Initial program 22.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f32 N/A

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

      2. lower-fabs.f32 93.6

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

   5. Applied rewrites 93.6%

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

   if 0.0399999991 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 59.3%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-+.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 5: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.03999999910593033)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ x (fabs x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (t_0 <= 0.03999999910593033f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((x + fabsf(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.03999999910593033))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(x + abs(x))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -2

   1. Initial program 50.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. lower--.f32 N/A

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

      16. lower-fabs.f32 96.3

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

   5. Applied rewrites 96.3%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0399999991

   1. Initial program 22.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f32 N/A

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

      2. lower-fabs.f32 93.6

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

   5. Applied rewrites 93.6%

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

   if 0.0399999991 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 59.3%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f32 N/A

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

      8. lower-fabs.f32 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 6: 21.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<=
      (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)
      0.03999999910593033)
   (copysign (- (/ (fabs x) x)) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= 0.03999999910593033f) {
		tmp = copysignf(-(fabsf(x) / x), x);
	} else {
		tmp = copysignf(logf(x), 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(0.03999999910593033))
		tmp = copysign(Float32(-Float32(abs(x) / x)), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if ((sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))))) <= single(0.03999999910593033))
		tmp = sign(x) * abs(-(abs(x) / x));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = 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) < 0.0399999991

   1. Initial program 30.9%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-fabs.f32 N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. log-rec N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-log.f32 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f32 18.9

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

   5. Applied rewrites 18.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 13.9%

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

   if 0.0399999991 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 59.3%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log.f32 43.9

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

   5. Applied rewrites 43.9%

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

Alternative 7: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.03999999910593033)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ x (fabs x))) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.03999999910593033f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((x + fabsf(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.03999999910593033))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(x + abs(x))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0399999991

   1. Initial program 30.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f32 N/A

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

      2. lower-fabs.f32 78.6

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

   5. Applied rewrites 78.6%

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

   if 0.0399999991 < x

   1. Initial program 59.3%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f32 N/A

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

      8. lower-fabs.f32 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 8: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (log1p (fabs x)) x))

C

float code(float x) {
	return copysignf(log1pf(fabsf(x)), x);
}

Julia

function code(x)
	return copysign(log1p(abs(x)), x)
end

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in x around 0

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

   1. lower-log1p.f32 N/A

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

   2. lower-fabs.f32 70.2

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

5. Applied rewrites 70.2%

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

Alternative 9: 16.0% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (copysign (- (/ (fabs x) x)) x))

C

float code(float x) {
	return copysignf(-(fabsf(x) / x), x);
}

Julia

function code(x)
	return copysign(Float32(-Float32(abs(x) / x)), x)
end

MATLAB

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in x around -inf

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-fabs.f32 N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-frac N/A

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

   11. distribute-neg-frac2 N/A

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

   12. mul-1-neg N/A

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

   13. log-rec N/A

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

   14. lower-neg.f32 N/A

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

   15. lower-log.f32 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f32 14.4

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

5. Applied rewrites 14.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 15.6%

   \[\leadsto \mathsf{copysign}\left(-\frac{\left|x\right|}{x}, x\right) \]
9. 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 2024219 
(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))