Rust f32::asinh

Percentage Accurate: 38.0% → 75.8%

Time: 8.1s

Alternatives: 11

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: 38.0% 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: 75.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

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 (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.4000000059604645)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (+ (fabs x) x) (/ 0.5 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((((-0.5f / x) - x) + fabsf(x))), x);
	} else if (t_0 <= 0.4000000059604645f) {
		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(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(Float32(Float32(Float32(-0.5) / x) - x) + abs(x))), x);
	elseif (t_0 <= Float32(0.4000000059604645))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(Float32(abs(x) + 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 43.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{-1 \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), x\right) \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.400000006

   1. Initial program 21.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.6%

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

   if 0.400000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 53.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. lower-+.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f32 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 97.5%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

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 (- (fabs x) x)) x)
     (if (<= t_0 0.4000000059604645)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (+ (fabs x) x) (/ 0.5 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((fabsf(x) - x)), x);
	} else if (t_0 <= 0.4000000059604645f) {
		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(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(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.4000000059604645))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(Float32(abs(x) + 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 43.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}, x\right) \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.400000006

   1. Initial program 21.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.6%

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

   if 0.400000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 53.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. lower-+.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f32 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 97.5%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.4000000059604645)
       (copysign (log1p (fabs x)) x)
       (copysign (log (/ 0.5 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((fabsf(x) - x)), x);
	} else if (t_0 <= 0.4000000059604645f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((0.5f / 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(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.4000000059604645))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 43.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}, x\right) \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.400000006

   1. Initial program 21.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.6%

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

   if 0.400000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 53.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. lower-+.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f32 N/A

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

      17. +-commutative N/A

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.4%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

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 (- (fabs x) x)) x)
     (if (<= t_0 0.4000000059604645)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (fabs 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((fabsf(x) - x)), x);
	} else if (t_0 <= 0.4000000059604645f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((fabsf(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(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.4000000059604645))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(abs(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 43.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}, x\right) \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.400000006

   1. Initial program 21.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.6%

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

   if 0.400000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 53.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 96.0

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

   5. Applied rewrites 96.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \end{array} \]

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 0.0) (copysign (log1p x) x) (copysign (log (+ 1.0 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 <= 0.0f) {
		tmp = copysignf(log1pf(x), x);
	} else {
		tmp = copysignf(logf((1.0f + 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(0.0))
		tmp = copysign(log1p(x), x);
	else
		tmp = copysign(log(Float32(Float32(1.0) + 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 43.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   if -2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0

   1. Initial program 16.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.3%

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

   if 0.0 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 54.6%

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

      1. lift-+.f32 N/A

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

      2. flip-+ N/A

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

      3. lift-*.f32 N/A

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

      4. difference-of-sqr-1 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. pow-prod-down N/A

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

      13. pow-prod-up N/A

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

      14. lower-pow.f32 N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 N/A

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

      18. lower--.f32 31.2

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. 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 27.1

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

   7. Applied rewrites 27.1%

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

   9. Applied rewrites 45.6%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 21.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\left|x\right|}{-x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<=
      (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)
      0.4000000059604645)
   (copysign (/ (fabs x) (- x)) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((sqrtf((1.0f + (x * x))) + fabsf(x))), x) <= 0.4000000059604645f) {
		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(sqrt(Float32(Float32(1.0) + Float32(x * x))) + abs(x))), x) <= Float32(0.4000000059604645))
		tmp = copysign(Float32(abs(x) / Float32(-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((sqrt((single(1.0) + (x * x))) + abs(x))))) <= single(0.4000000059604645))
		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) < 0.400000006

   1. Initial program 27.5%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot \log \left(\frac{-1}{x}\right) + -1 \cdot \frac{\left|x\right|}{x}}, x\right) \]
   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. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f32 N/A

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

      9. lower-fabs.f32 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-log.f32 N/A

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

      13. lower-/.f32 17.3

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

   5. Applied rewrites 17.3%

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

   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 13.3%

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

   if 0.400000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 53.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}, x\right) \]
   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 44.2

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

   5. Applied rewrites 44.2%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\left|x\right|}{-x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -20.0)
   (copysign (log (- x)) x)
   (if (<= x 0.4000000059604645)
     (copysign (log1p (fabs x)) x)
     (copysign (log (+ (fabs x) x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -20.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-20.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(abs(x) + x)), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -20

   1. Initial program 43.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   if -20 < x < 0.400000006

   1. Initial program 21.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.6%

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

   if 0.400000006 < x

   1. Initial program 53.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 96.0

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

   5. Applied rewrites 96.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -20.0)
   (copysign (log (- x)) x)
   (if (<= x 0.4000000059604645)
     (copysign (log1p (fabs x)) x)
     (copysign (log (+ 1.0 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -20.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((1.0f + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-20.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(Float32(1.0) + x)), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -20

   1. Initial program 43.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   if -20 < x < 0.400000006

   1. Initial program 21.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.6%

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

   if 0.400000006 < x

   1. Initial program 53.2%

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

      1. lift-+.f32 N/A

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

      2. flip-+ N/A

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

      3. lift-*.f32 N/A

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

      4. difference-of-sqr-1 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. metadata-eval N/A

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

      9. lower--.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. pow-prod-down N/A

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

      13. pow-prod-up N/A

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

      14. lower-pow.f32 N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. lower-+.f32 N/A

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

      18. lower--.f32 24.5

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

   4. Applied rewrites 24.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. 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 11.3

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

   7. Applied rewrites 11.3%

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

   9. Applied rewrites 44.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -20.0)
   (copysign (log (- x)) x)
   (if (<= x 5.000000229068525e-19)
     (copysign (log1p x) x)
     (copysign (log x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -20.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 5.000000229068525e-19f) {
		tmp = copysignf(log1pf(x), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-20.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(5.000000229068525e-19))
		tmp = copysign(log1p(x), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -20

   1. Initial program 43.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   if -20 < x < 5.00000023e-19

   1. Initial program 18.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   7. Applied rewrites 97.9%

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

   9. Applied rewrites 97.9%

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

   if 5.00000023e-19 < x

   1. Initial program 44.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}, x\right) \]
   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 32.8

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

   5. Applied rewrites 32.8%

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

Alternative 10: 30.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 5.000000229068525e-19) (copysign (log1p x) x) (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 5.000000229068525e-19f) {
		tmp = copysignf(log1pf(x), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000229068525e-19))
		tmp = copysign(log1p(x), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000023e-19

   1. Initial program 27.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   7. Applied rewrites 66.8%

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

   9. Applied rewrites 66.8%

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

   if 5.00000023e-19 < x

   1. Initial program 44.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}, x\right) \]
   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 32.8

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

   5. Applied rewrites 32.8%

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

Alternative 11: 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(abs(x) / Float32(-x)), x)
end

MATLAB

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

Derivation

1. Initial program 33.7%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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. distribute-neg-frac2 N/A

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

   7. mul-1-neg N/A

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

   8. lower-/.f32 N/A

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

   9. lower-fabs.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. lower-log.f32 N/A

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

   13. lower-/.f32 13.1

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

5. Applied rewrites 13.1%

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

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 15.2%

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

Developer Target 1: 53.4% 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 2024244 
(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))