Rust f32::asinh

Percentage Accurate: 37.7% → 72.0%

Time: 7.3s

Alternatives: 11

Speedup: 0.3×

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: 37.7% 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: 72.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((sqrtf((1.0f + (x * x))) + fabsf(x))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(logf((((-0.5f / x) - x) + fabsf(x))), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf(((x - (-0.5f / x)) + fabsf(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(sqrt(Float32(Float32(1.0) + Float32(x * x))) + abs(x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(Float32(Float32(-0.5) / x) - x) + abs(x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(Float32(x - Float32(Float32(-0.5) / 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) < -1

   1. Initial program 49.6%

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

   3. Taylor expanded in x around -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-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

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

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

      14. lower--.f32 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f32 97.9

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

   5. Applied rewrites 97.9%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 23.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.3%

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

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.2%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

      13. lft-mult-inverse N/A

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

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

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

      15. lower--.f32 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f32 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((sqrtf((1.0f + (x * x))) + fabsf(x))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf(((x - (-0.5f / x)) + fabsf(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(sqrt(Float32(Float32(1.0) + Float32(x * x))) + abs(x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(Float32(x - Float32(Float32(-0.5) / 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) < -1

   1. Initial program 49.6%

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

   3. Taylor expanded in x around -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-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f32 N/A

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

      17. lower-fabs.f32 97.2

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

   5. Applied rewrites 97.2%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 23.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.3%

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

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.2%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

      13. lft-mult-inverse N/A

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

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

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

      15. lower--.f32 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f32 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ x x)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((sqrtf((1.0f + (x * x))) + fabsf(x))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(sqrt(Float32(Float32(1.0) + Float32(x * x))) + abs(x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(x + 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) < -1

   1. Initial program 49.6%

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

   3. Taylor expanded in x around -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-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f32 N/A

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

      17. lower-fabs.f32 97.2

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

   5. Applied rewrites 97.2%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 23.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.3%

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

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f32 N/A

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

      9. lower-fabs.f32 97.1

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

   5. Applied rewrites 97.1%

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

   7. Applied rewrites 97.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (+ 1.0 (fabs x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ x x)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((sqrtf((1.0f + (x * x))) + fabsf(x))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(logf((1.0f + fabsf(x))), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(sqrt(Float32(Float32(1.0) + Float32(x * x))) + abs(x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(1.0) + abs(x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(x + 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) < -1

   1. Initial program 49.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-fabs.f32 44.6

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

   5. Applied rewrites 44.6%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 23.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.3%

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

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f32 N/A

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

      9. lower-fabs.f32 97.1

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

   5. Applied rewrites 97.1%

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

   7. Applied rewrites 97.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (- x)) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ x x)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((sqrtf((1.0f + (x * x))) + fabsf(x))), x);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(sqrt(Float32(Float32(1.0) + Float32(x * x))) + abs(x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(x + 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 48.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 44.8

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

   5. Applied rewrites 44.8%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 24.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.7%

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

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f32 N/A

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

      9. lower-fabs.f32 97.1

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

   5. Applied rewrites 97.1%

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

   7. Applied rewrites 97.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = copysign(log(Float32(sqrt(Float32(Float32(1.0) + Float32(x * x))) + abs(x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (t_0 <= Float32(2.0))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(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 48.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 44.8

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

   5. Applied rewrites 44.8%

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

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

   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

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   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 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.2%

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

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

   1. Initial program 55.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 44.1

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

   5. Applied rewrites 44.1%

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

4. Final simplification 45.9%

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

Alternative 7: 36.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 32.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.8%

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

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

   1. Initial program 55.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 44.1

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

   5. Applied rewrites 44.1%

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

4. Final simplification 53.8%

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

Alternative 8: 18.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, x \cdot x, \mathsf{fma}\left(0.3333333333333333, \left|x\right|, -0.5\right)\right) \cdot x, x, \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 4.0)
   (copysign
    (fma
     (* (fma -0.25 (* x x) (fma 0.3333333333333333 (fabs x) -0.5)) x)
     x
     (fabs x))
    x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 4.0f) {
		tmp = copysignf(fmaf((fmaf(-0.25f, (x * x), fmaf(0.3333333333333333f, fabsf(x), -0.5f)) * x), x, fabsf(x)), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(4.0))
		tmp = copysign(fma(Float32(fma(Float32(-0.25), Float32(x * x), fma(Float32(0.3333333333333333), abs(x), Float32(-0.5))) * x), x, abs(x)), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 32.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 67.8

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.8%

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

   9. Applied rewrites 40.2%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 67.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, x \cdot x, \mathsf{fma}\left(0.3333333333333333, \left|x\right|, -0.5\right)\right) \cdot x, \color{blue}{x}, \left|x\right|\right), x\right) \]

   if 4 < x

   1. Initial program 55.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 44.1

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

   5. Applied rewrites 44.1%

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

Alternative 9: 10.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, x \cdot x, \mathsf{fma}\left(0.3333333333333333, \left|x\right|, -0.5\right)\right) \cdot x, x, \left|x\right|\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign
  (fma
   (* (fma -0.25 (* x x) (fma 0.3333333333333333 (fabs x) -0.5)) x)
   x
   (fabs x))
  x))

C

float code(float x) {
	return copysignf(fmaf((fmaf(-0.25f, (x * x), fmaf(0.3333333333333333f, fabsf(x), -0.5f)) * x), x, fabsf(x)), x);
}

Julia

function code(x)
	return copysign(fma(Float32(fma(Float32(-0.25), Float32(x * x), fma(Float32(0.3333333333333333), abs(x), Float32(-0.5))) * x), x, abs(x)), x)
end

Derivation

1. Initial program 38.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 53.9

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

5. Applied rewrites 53.9%

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

7. Applied rewrites 53.9%

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

9. Applied rewrites 33.0%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 53.9%

    \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, x \cdot x, \mathsf{fma}\left(0.3333333333333333, \left|x\right|, -0.5\right)\right) \cdot x, \color{blue}{x}, \left|x\right|\right), x\right) \]
13. Add Preprocessing

Alternative 10: 10.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \left|x\right|, -0.5\right) \cdot x, x, \left|x\right|\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (fma (* (fma 0.3333333333333333 (fabs x) -0.5) x) x (fabs x)) x))

C

float code(float x) {
	return copysignf(fmaf((fmaf(0.3333333333333333f, fabsf(x), -0.5f) * x), x, fabsf(x)), x);
}

Julia

function code(x)
	return copysign(fma(Float32(fma(Float32(0.3333333333333333), abs(x), Float32(-0.5)) * x), x, abs(x)), x)
end

Derivation

1. Initial program 38.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 53.9

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

5. Applied rewrites 53.9%

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

7. Applied rewrites 53.9%

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

9. Applied rewrites 33.0%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 53.9%

    \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \left|x\right|, -0.5\right) \cdot x, \color{blue}{x}, \left|x\right|\right), x\right) \]
13. Add Preprocessing

Alternative 11: 10.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{fma}\left(-0.5, x \cdot x, \left|x\right|\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (fma -0.5 (* x x) (fabs x)) x))

C

float code(float x) {
	return copysignf(fmaf(-0.5f, (x * x), fabsf(x)), x);
}

Julia

function code(x)
	return copysign(fma(Float32(-0.5), Float32(x * x), abs(x)), x)
end

Derivation

1. Initial program 38.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 53.9

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

5. Applied rewrites 53.9%

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

7. Applied rewrites 53.9%

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

9. Applied rewrites 33.0%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 53.9%

    \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, \left|x\right|\right), x\right) \]
13. Add Preprocessing

Developer Target 1: 54.6% 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 2024331 
(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))