Rust f32::asinh

Percentage Accurate: 37.5% → 97.7%

Time: 5.8s

Alternatives: 10

Speedup: 2.2×

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: 37.5% 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: 97.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -0.5)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (- 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 <= -0.5f) {
		tmp = copysignf(logf((((-0.5f / x) - x) + fabsf(x))), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(-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(-0.5))
		tmp = copysign(log(Float32(Float32(Float32(Float32(-0.5) / x) - x) + abs(x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-x), x);
	else
		tmp = copysign(log(Float32(Float32(x - Float32(Float32(-0.5) / x)) + abs(x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((sqrt((single(1.0) + (x * x))) + abs(x))));
	tmp = single(0.0);
	if (t_0 <= single(-0.5))
		tmp = sign(x) * abs(log((((single(-0.5) / x) - x) + abs(x))));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = sign(x) * abs(-x);
	else
		tmp = sign(x) * abs(log(((x - (single(-0.5) / x)) + abs(x))));
	end
	tmp_2 = 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) < -0.5

   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

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{-1 \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), x\right) \]
   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 96.5

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

   5. Applied rewrites 96.5%

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

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f32 N/A

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

      2. lower-fabs.f32 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

   9. Applied rewrites 19.2%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.0%

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

   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 61.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}\right), x\right) \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.3%

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

Alternative 2: 97.3% accurate, 0.3× speedup

Math

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

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 (- 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(-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(Float32(-x), x);
	else
		tmp = copysign(log(Float32(Float32(x - Float32(Float32(-0.5) / x)) + abs(x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((sqrt((single(1.0) + (x * x))) + abs(x))));
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = sign(x) * abs(log((abs(x) - x)));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = sign(x) * abs(-x);
	else
		tmp = sign(x) * abs(log(((x - (single(-0.5) / x)) + abs(x))));
	end
	tmp_2 = 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 54.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}, x\right) \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

   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 22.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 97.4%

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

   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 61.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}\right), x\right) \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.9%

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

Alternative 3: 97.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (- 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 <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(-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(-1.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-x), x);
	else
		tmp = copysign(log(Float32(abs(x) + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((sqrt((single(1.0) + (x * x))) + abs(x))));
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = sign(x) * abs(log((abs(x) - x)));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = sign(x) * abs(-x);
	else
		tmp = sign(x) * abs(log((abs(x) + x)));
	end
	tmp_2 = 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 54.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}, x\right) \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

   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 22.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 97.4%

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

   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 61.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 97.7%

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

Alternative 4: 62.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (copysignf(logf((sqrtf((1.0f + (x * x))) + fabsf(x))), x) <= 0.20000000298023224f) {
		tmp = copysignf(-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(0.20000000298023224))
		tmp = copysign(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(0.20000000298023224))
		tmp = sign(x) * abs(-x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 33.9%

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

   3. 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 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 67.2%

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

   9. Applied rewrites 28.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 67.2%

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

   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 61.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}, x\right) \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

4. Final simplification 59.6%

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

Alternative 5: 57.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (copysignf(logf((sqrtf((1.0f + (x * x))) + fabsf(x))), x) <= 0.20000000298023224f) {
		tmp = copysignf(-x, x);
	} else {
		tmp = copysignf((fabsf(x) / -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(0.20000000298023224))
		tmp = copysign(Float32(-x), x);
	else
		tmp = copysign(Float32(abs(x) / Float32(-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(0.20000000298023224))
		tmp = sign(x) * abs(-x);
	else
		tmp = sign(x) * abs((abs(x) / -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) < 0.200000003

   1. Initial program 33.9%

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

   3. 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 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 67.2%

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

   9. Applied rewrites 28.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 67.2%

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

   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 61.9%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. mul-1-neg N/A

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

      6. 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 -0.0

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

   5. Applied rewrites -0.0%

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

   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 20.9%

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

4. Final simplification 51.6%

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

Alternative 6: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.5)
   (copysign (log (- 1.0 x)) x)
   (if (<= x 0.20000000298023224)
     (copysign (- x) x)
     (copysign (log (+ (fabs x) x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.5f) {
		tmp = copysignf(logf((1.0f - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-x, x);
	} else {
		tmp = copysignf(logf((fabsf(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.5))
		tmp = copysign(log(Float32(Float32(1.0) - x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-x), x);
	else
		tmp = copysign(log(Float32(abs(x) + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.5))
		tmp = sign(x) * abs(log((single(1.0) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(-x);
	else
		tmp = sign(x) * abs(log((abs(x) + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 54.7%

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

   3. Taylor expanded in x around 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 11.8

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

   5. Applied rewrites 11.8%

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

   7. Applied rewrites 11.8%

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

   9. Applied rewrites 43.8%

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

   if -1.5 < x < 0.200000003

   1. Initial program 22.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 97.4%

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

   if 0.200000003 < x

   1. Initial program 61.9%

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

   3. 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 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 7: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.5)
   (copysign (log (- 1.0 x)) x)
   (if (<= x 0.20000000298023224)
     (copysign (- x) x)
     (copysign (log (+ 1.0 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.5f) {
		tmp = copysignf(logf((1.0f - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-x, x);
	} else {
		tmp = copysignf(logf((1.0f + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.5))
		tmp = copysign(log(Float32(Float32(1.0) - x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-x), x);
	else
		tmp = copysign(log(Float32(Float32(1.0) + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.5))
		tmp = sign(x) * abs(log((single(1.0) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(-x);
	else
		tmp = sign(x) * abs(log((single(1.0) + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 54.7%

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

   3. Taylor expanded in x around 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 11.8

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

   5. Applied rewrites 11.8%

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

   7. Applied rewrites 11.8%

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

   9. Applied rewrites 43.8%

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

   if -1.5 < x < 0.200000003

   1. Initial program 22.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 97.4%

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

   if 0.200000003 < x

   1. Initial program 61.9%

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

   3. 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 11.1

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

   5. Applied rewrites 11.1%

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

   7. Applied rewrites 11.1%

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

   9. Applied rewrites 11.1%

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

   11. Applied rewrites 44.4%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (- x)) x)
   (if (<= x 0.20000000298023224)
     (copysign (- x) x)
     (copysign (log (+ 1.0 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-x, x);
	} else {
		tmp = copysignf(logf((1.0f + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-x), x);
	else
		tmp = copysign(log(Float32(Float32(1.0) + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(-x);
	else
		tmp = sign(x) * abs(log((single(1.0) + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 54.0%

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

   3. Taylor expanded in x around -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 44.0

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

   5. Applied rewrites 44.0%

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

   if -10 < x < 0.200000003

   1. Initial program 23.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 96.8

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.8%

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

   9. Applied rewrites 19.4%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 96.8%

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

   if 0.200000003 < x

   1. Initial program 61.9%

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

   3. 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 11.1

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

   5. Applied rewrites 11.1%

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

   7. Applied rewrites 11.1%

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

   9. Applied rewrites 11.1%

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

   11. Applied rewrites 44.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (- x)) x)
   (if (<= x 0.20000000298023224) (copysign (- x) x) (copysign (log x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(-x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(-x), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(-x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 54.0%

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

   3. Taylor expanded in x around -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 44.0

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

   5. Applied rewrites 44.0%

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

   if -10 < x < 0.200000003

   1. Initial program 23.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 96.8

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.8%

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

   9. Applied rewrites 19.4%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 96.8%

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

   if 0.200000003 < x

   1. Initial program 61.9%

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

   3. 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 44.4

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

   5. Applied rewrites 44.4%

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

Alternative 10: 54.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 43.3%

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

3. Taylor expanded in x around 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 48.4

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

5. Applied rewrites 48.4%

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

7. Applied rewrites 48.4%

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

9. Applied rewrites 18.6%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 48.4%

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

Developer Target 1: 54.4% 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 2024285 
(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))