Rust f32::asinh

Percentage Accurate: 37.5% → 98.6%

Time: 12.5s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.5% 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: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.001388888888888889\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_2 := \left|x\right| + 1\\ t_3 := t\_2 \cdot t\_2\\ \mathbf{if}\;t\_1 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{t\_2} + \frac{1}{t\_3}, -0.125 + t\_0 \cdot 45, \frac{t\_0 \cdot 30}{t\_2 \cdot t\_3}\right), \frac{0.5}{t\_2}\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (* (* x x) 0.001388888888888889))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_2 (+ (fabs x) 1.0))
        (t_3 (* t_2 t_2)))
   (if (<= t_1 -2.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_1 0.20000000298023224)
       (copysign
        (fma
         (* x x)
         (fma
          (* x x)
          (fma
           (+ (/ 1.0 t_2) (/ 1.0 t_3))
           (+ -0.125 (* t_0 45.0))
           (/ (* t_0 30.0) (* t_2 t_3)))
          (/ 0.5 t_2))
         (log1p (fabs x)))
        x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = (x * x) * 0.001388888888888889f;
	float t_1 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_2 = fabsf(x) + 1.0f;
	float t_3 = t_2 * t_2;
	float tmp;
	if (t_1 <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (t_1 <= 0.20000000298023224f) {
		tmp = copysignf(fmaf((x * x), fmaf((x * x), fmaf(((1.0f / t_2) + (1.0f / t_3)), (-0.125f + (t_0 * 45.0f)), ((t_0 * 30.0f) / (t_2 * t_3))), (0.5f / t_2)), log1pf(fabsf(x))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(Float32(x * x) * Float32(0.001388888888888889))
	t_1 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_2 = Float32(abs(x) + Float32(1.0))
	t_3 = Float32(t_2 * t_2)
	tmp = Float32(0.0)
	if (t_1 <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (t_1 <= Float32(0.20000000298023224))
		tmp = copysign(fma(Float32(x * x), fma(Float32(x * x), fma(Float32(Float32(Float32(1.0) / t_2) + Float32(Float32(1.0) / t_3)), Float32(Float32(-0.125) + Float32(t_0 * Float32(45.0))), Float32(Float32(t_0 * Float32(30.0)) / Float32(t_2 * t_3))), Float32(Float32(0.5) / t_2)), log1p(abs(x))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.0%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.6%

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

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

   1. Initial program 24.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{24} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) + \frac{1}{720} \cdot \left({x}^{2} \cdot \left(45 \cdot \frac{1}{1 + \left|x\right|} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}}\right)\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)}, x\right) \]

   5. Simplified 99.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{1 + \left|x\right|} + \frac{1}{\left(1 + \left|x\right|\right) \cdot \left(1 + \left|x\right|\right)}, -0.125 + \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot 45, \frac{\left(\left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot 30}{\left(1 + \left|x\right|\right) \cdot \left(\left(1 + \left|x\right|\right) \cdot \left(1 + \left|x\right|\right)\right)}\right), \frac{0.5}{1 + \left|x\right|}\right), \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

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

   1. Initial program 60.1%

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

   3. Taylor expanded in x around 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. associate-+r+ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. +-lowering-+.f32 N/A

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

      12. fabs-lowering-fabs.f32 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. associate-*l/ N/A

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

      19. lft-mult-inverse N/A

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

   5. Simplified 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{\left|x\right| + 1} + \frac{1}{\left(\left|x\right| + 1\right) \cdot \left(\left|x\right| + 1\right)}, -0.125 + \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot 45, \frac{\left(\left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot 30}{\left(\left|x\right| + 1\right) \cdot \left(\left(\left|x\right| + 1\right) \cdot \left(\left|x\right| + 1\right)\right)}\right), \frac{0.5}{\left|x\right| + 1}\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x \cdot x, \frac{\left(x \cdot x\right) \cdot -0.125}{\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, 0.5, 1\right), 1\right)}, \mathsf{log1p}\left(\mathsf{fma}\left(0.5, x \cdot x, \left|x\right|\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (fma
         (* x x)
         (/ (* (* x x) -0.125) (fma (fabs x) (fma (fabs x) 0.5 1.0) 1.0))
         (log1p (fma 0.5 (* x x) (fabs x))))
        x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(fmaf((x * x), (((x * x) * -0.125f) / fmaf(fabsf(x), fmaf(fabsf(x), 0.5f, 1.0f), 1.0f)), log1pf(fmaf(0.5f, (x * x), fabsf(x)))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(fma(Float32(x * x), Float32(Float32(Float32(x * x) * Float32(-0.125)) / fma(abs(x), fma(abs(x), Float32(0.5), Float32(1.0)), Float32(1.0))), log1p(fma(Float32(0.5), Float32(x * x), abs(x)))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.0%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.6%

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

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

   1. Initial program 24.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right) + 1\right)}, x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + \left|x\right|\right)} + 1\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + \left|x\right|\right) + 1\right), x\right) \]

      4. sqr-abs N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + \left|x\right|\right) + 1\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left|x\right| \cdot \left(\left|x\right| \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)} + \left|x\right|\right) + 1\right), x\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right) + \color{blue}{\left|x\right| \cdot 1}\right) + 1\right), x\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left|x\right| \cdot \left(\left|x\right| \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + 1\right)} + 1\right), x\right) \]

      8. accelerator-lowering-fma.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right|, \left|x\right| \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + 1, 1\right)\right)}, x\right) \]

      9. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\color{blue}{\left|x\right|}, \left|x\right| \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + 1, 1\right)\right), x\right) \]

      10. accelerator-lowering-fma.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\mathsf{fma}\left(\left|x\right|, \frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}, 1\right)}, 1\right)\right), x\right) \]

      11. fabs-lowering-fabs.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\color{blue}{\left|x\right|}, \frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}, 1\right), 1\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \color{blue}{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{2}}, 1\right), 1\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{2} \cdot \frac{-1}{8}} + \frac{1}{2}, 1\right), 1\right)\right), x\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{8} + \frac{1}{2}, 1\right), 1\right)\right), x\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \color{blue}{x \cdot \left(x \cdot \frac{-1}{8}\right)} + \frac{1}{2}, 1\right), 1\right)\right), x\right) \]

      16. accelerator-lowering-fma.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{8}, \frac{1}{2}\right)}, 1\right), 1\right)\right), x\right) \]

      17. *-lowering-*.f32 23.8

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

   5. Simplified 23.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)\right) + \frac{-1}{8} \cdot \frac{{x}^{2} \cdot {\left(\left|x\right|\right)}^{2}}{1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)}}, x\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{-1}{8} \cdot \frac{{x}^{2} \cdot {\left(\left|x\right|\right)}^{2}}{1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)} + \log \left(1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)\right)}, x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{x}^{2} \cdot {\left(\left|x\right|\right)}^{2}}{1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)} \cdot \frac{-1}{8}} + \log \left(1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)\right), x\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left({x}^{2} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)}\right)} \cdot \frac{-1}{8} + \log \left(1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)\right), x\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{2} \cdot \left(\frac{{\left(\left|x\right|\right)}^{2}}{1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)} \cdot \frac{-1}{8}\right)} + \log \left(1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)}\right)} + \log \left(1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)\right), x\right) \]

      6. accelerator-lowering-fma.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)}, \log \left(1 + \left|x\right| \cdot \left(1 + \frac{1}{2} \cdot \left|x\right|\right)\right)\right)}, x\right) \]

   8. Simplified 99.1%

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

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

   1. Initial program 60.1%

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

   3. Taylor expanded in x around 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. associate-+r+ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. +-lowering-+.f32 N/A

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

      12. fabs-lowering-fabs.f32 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. associate-*l/ N/A

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

      19. lft-mult-inverse N/A

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

   5. Simplified 99.1%

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

Alternative 3: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot -0.125, 0.5\right), x \cdot x, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (log1p (fma (fma x (* x -0.125) 0.5) (* x x) (fabs x))) x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(log1pf(fmaf(fmaf(x, (x * -0.125f), 0.5f), (x * x), fabsf(x))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(fma(fma(x, Float32(x * Float32(-0.125)), Float32(0.5)), Float32(x * x), abs(x))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.0%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.6%

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

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

   1. Initial program 24.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right) + 1\right)}, x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + \left|x\right|\right)} + 1\right), x\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + \left|x\right|\right) + 1\right), x\right) \]

      4. sqr-abs N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + \left|x\right|\right) + 1\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left|x\right| \cdot \left(\left|x\right| \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)} + \left|x\right|\right) + 1\right), x\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right) + \color{blue}{\left|x\right| \cdot 1}\right) + 1\right), x\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left|x\right| \cdot \left(\left|x\right| \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + 1\right)} + 1\right), x\right) \]

      8. accelerator-lowering-fma.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right|, \left|x\right| \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + 1, 1\right)\right)}, x\right) \]

      9. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\color{blue}{\left|x\right|}, \left|x\right| \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) + 1, 1\right)\right), x\right) \]

      10. accelerator-lowering-fma.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\mathsf{fma}\left(\left|x\right|, \frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}, 1\right)}, 1\right)\right), x\right) \]

      11. fabs-lowering-fabs.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\color{blue}{\left|x\right|}, \frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}, 1\right), 1\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \color{blue}{\frac{-1}{8} \cdot {x}^{2} + \frac{1}{2}}, 1\right), 1\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{2} \cdot \frac{-1}{8}} + \frac{1}{2}, 1\right), 1\right)\right), x\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{8} + \frac{1}{2}, 1\right), 1\right)\right), x\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \color{blue}{x \cdot \left(x \cdot \frac{-1}{8}\right)} + \frac{1}{2}, 1\right), 1\right)\right), x\right) \]

      16. accelerator-lowering-fma.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{8}, \frac{1}{2}\right)}, 1\right), 1\right)\right), x\right) \]

      17. *-lowering-*.f32 23.8

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

   5. Simplified 23.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x, x \cdot -0.125, 0.5\right), 1\right), 1\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot \left(x \cdot \frac{-1}{8}\right) + \frac{1}{2}\right) + 1\right)\right)}, x\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot \left(x \cdot \frac{-1}{8}\right) + \frac{1}{2}\right) + 1\right)\right)}, x\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot \left(x \cdot \frac{-1}{8}\right) + \frac{1}{2}\right)\right) + \left|x\right| \cdot 1}\right), x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| \cdot \left(x \cdot \left(x \cdot \frac{-1}{8}\right) + \frac{1}{2}\right)\right) \cdot \left|x\right|} + \left|x\right| \cdot 1\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left(x \cdot \left(x \cdot \frac{-1}{8}\right) + \frac{1}{2}\right) \cdot \left|x\right|\right)} \cdot \left|x\right| + \left|x\right| \cdot 1\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{8}\right) + \frac{1}{2}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)} + \left|x\right| \cdot 1\right), x\right) \]

      7. sqr-abs N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x \cdot \left(x \cdot \frac{-1}{8}\right) + \frac{1}{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left|x\right| \cdot 1\right), x\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x \cdot \left(x \cdot \frac{-1}{8}\right) + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left|x\right|}\right), x\right) \]

      9. accelerator-lowering-fma.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \frac{-1}{8}\right) + \frac{1}{2}, x \cdot x, \left|x\right|\right)}\right), x\right) \]

      10. accelerator-lowering-fma.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{8}, \frac{1}{2}\right)}, x \cdot x, \left|x\right|\right)\right), x\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{8}}, \frac{1}{2}\right), x \cdot x, \left|x\right|\right)\right), x\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{-1}{8}, \frac{1}{2}\right), \color{blue}{x \cdot x}, \left|x\right|\right)\right), x\right) \]

      13. fabs-lowering-fabs.f32 99.1

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

   7. Applied egg-rr 99.1%

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

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

   1. Initial program 60.1%

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

   3. Taylor expanded in x around 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. associate-+r+ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. +-lowering-+.f32 N/A

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

      12. fabs-lowering-fabs.f32 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. associate-*l/ N/A

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

      19. lft-mult-inverse N/A

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

   5. Simplified 99.1%

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

Alternative 4: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (log1p (fma x (* x 0.5) (fabs x))) x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(log1pf(fmaf(x, (x * 0.5f), fabsf(x))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(fma(x, Float32(x * Float32(0.5)), abs(x))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.0%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.6%

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

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

   1. Initial program 24.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left(\color{blue}{\left(\frac{-1}{6} \cdot -3\right)} \cdot \left|x\right|\right) \cdot \left|x\right| + \left|x\right|\right) + 1\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(\frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right)\right)} \cdot \left|x\right| + \left|x\right|\right) + 1\right), x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left(\frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right)\right) \cdot \left|x\right| + \color{blue}{1 \cdot \left|x\right|}\right) + 1\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right) + 1\right)} + 1\right), x\right) \]

      10. accelerator-lowering-fma.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right|, \frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right) + 1, 1\right)\right)}, x\right) \]

      11. fabs-lowering-fabs.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\color{blue}{\left|x\right|}, \frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right) + 1, 1\right)\right), x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\left(-3 \cdot \left|x\right|\right) \cdot \frac{-1}{6}} + 1, 1\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\left(\left|x\right| \cdot -3\right)} \cdot \frac{-1}{6} + 1, 1\right)\right), x\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\left|x\right| \cdot \left(-3 \cdot \frac{-1}{6}\right)} + 1, 1\right)\right), x\right) \]

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f32 N/A

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

      17. fabs-lowering-fabs.f32 23.0

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

   5. Simplified 23.0%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. sqr-abs N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

      9. *-lowering-*.f32 N/A

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

      10. fabs-lowering-fabs.f32 97.9

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

   7. Applied egg-rr 97.9%

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

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

   1. Initial program 60.1%

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

   3. Taylor expanded in x around 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. associate-+r+ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. +-lowering-+.f32 N/A

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

      12. fabs-lowering-fabs.f32 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. associate-*l/ N/A

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

      19. lft-mult-inverse N/A

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

   5. Simplified 99.1%

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

Alternative 5: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (log1p (fma x (* x 0.5) (fabs x))) x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(log1pf(fmaf(x, (x * 0.5f), fabsf(x))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(fma(x, Float32(x * Float32(0.5)), abs(x))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.0%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. --lowering--.f32 N/A

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

      16. fabs-lowering-fabs.f32 98.3

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

   5. Simplified 98.3%

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

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

   1. Initial program 24.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left(\color{blue}{\left(\frac{-1}{6} \cdot -3\right)} \cdot \left|x\right|\right) \cdot \left|x\right| + \left|x\right|\right) + 1\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(\frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right)\right)} \cdot \left|x\right| + \left|x\right|\right) + 1\right), x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left(\frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right)\right) \cdot \left|x\right| + \color{blue}{1 \cdot \left|x\right|}\right) + 1\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right) + 1\right)} + 1\right), x\right) \]

      10. accelerator-lowering-fma.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right|, \frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right) + 1, 1\right)\right)}, x\right) \]

      11. fabs-lowering-fabs.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\color{blue}{\left|x\right|}, \frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right) + 1, 1\right)\right), x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\left(-3 \cdot \left|x\right|\right) \cdot \frac{-1}{6}} + 1, 1\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\left(\left|x\right| \cdot -3\right)} \cdot \frac{-1}{6} + 1, 1\right)\right), x\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\left|x\right| \cdot \left(-3 \cdot \frac{-1}{6}\right)} + 1, 1\right)\right), x\right) \]

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f32 N/A

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

      17. fabs-lowering-fabs.f32 23.0

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

   5. Simplified 23.0%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. sqr-abs N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

      9. *-lowering-*.f32 N/A

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

      10. fabs-lowering-fabs.f32 97.9

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

   7. Applied egg-rr 97.9%

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

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

   1. Initial program 60.1%

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

   3. Taylor expanded in x around 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. associate-+r+ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. +-lowering-+.f32 N/A

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

      12. fabs-lowering-fabs.f32 N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. associate-*l/ N/A

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

      19. lft-mult-inverse N/A

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

   5. Simplified 99.1%

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

Alternative 6: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (log1p (fma x (* x 0.5) (fabs x))) x)
       (copysign (log (+ x (fabs x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(log1pf(fmaf(x, (x * 0.5f), fabsf(x))), x);
	} else {
		tmp = copysignf(logf((x + fabsf(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(fma(x, Float32(x * Float32(0.5)), abs(x))), x);
	else
		tmp = copysign(log(Float32(x + abs(x))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.0%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. --lowering--.f32 N/A

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

      16. fabs-lowering-fabs.f32 98.3

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

   5. Simplified 98.3%

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

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

   1. Initial program 24.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left(\color{blue}{\left(\frac{-1}{6} \cdot -3\right)} \cdot \left|x\right|\right) \cdot \left|x\right| + \left|x\right|\right) + 1\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(\frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right)\right)} \cdot \left|x\right| + \left|x\right|\right) + 1\right), x\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left(\frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right)\right) \cdot \left|x\right| + \color{blue}{1 \cdot \left|x\right|}\right) + 1\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left|x\right| \cdot \left(\frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right) + 1\right)} + 1\right), x\right) \]

      10. accelerator-lowering-fma.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right|, \frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right) + 1, 1\right)\right)}, x\right) \]

      11. fabs-lowering-fabs.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\color{blue}{\left|x\right|}, \frac{-1}{6} \cdot \left(-3 \cdot \left|x\right|\right) + 1, 1\right)\right), x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\left(-3 \cdot \left|x\right|\right) \cdot \frac{-1}{6}} + 1, 1\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\left(\left|x\right| \cdot -3\right)} \cdot \frac{-1}{6} + 1, 1\right)\right), x\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right|, \color{blue}{\left|x\right| \cdot \left(-3 \cdot \frac{-1}{6}\right)} + 1, 1\right)\right), x\right) \]

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f32 N/A

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

      17. fabs-lowering-fabs.f32 23.0

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

   5. Simplified 23.0%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. sqr-abs N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

      9. *-lowering-*.f32 N/A

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

      10. fabs-lowering-fabs.f32 97.9

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

   7. Applied egg-rr 97.9%

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

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

   1. Initial program 60.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.0%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. --lowering--.f32 N/A

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

      16. fabs-lowering-fabs.f32 98.3

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

   5. Simplified 98.3%

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

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

   1. Initial program 24.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. accelerator-lowering-log1p.f32 N/A

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

      2. fabs-lowering-fabs.f32 92.5

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

   5. Simplified 92.5%

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

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

   1. Initial program 60.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.6%

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

4. Final simplification 95.0%

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

Alternative 8: 82.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x) <= Float32(0.20000000298023224))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(x + abs(x))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 31.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. accelerator-lowering-log1p.f32 N/A

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

      2. fabs-lowering-fabs.f32 78.3

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

   5. Simplified 78.3%

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

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

   1. Initial program 60.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.6%

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

4. Final simplification 82.9%

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

Alternative 9: 27.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000001076526666 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.000001076526666e-40)
   (copysign (log (- x)) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -5.000001076526666e-40f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.000001076526666e-40))
		tmp = copysign(log(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(-5.000001076526666e-40))
		tmp = sign(x) * abs(log(-x));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.0000011e-40

   1. Initial program 37.6%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
   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. neg-lowering-neg.f32 27.0

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

   5. Simplified 27.0%

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

   if -5.0000011e-40 < x

   1. Initial program 39.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. log-lowering-log.f32 25.3

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

   5. Simplified 25.3%

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

Alternative 10: 20.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot \frac{1}{\left|x\right| + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.10000000149011612)
   (copysign (* (* x 0.5) (* x (/ 1.0 (+ (fabs x) 1.0)))) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.10000000149011612f) {
		tmp = copysignf(((x * 0.5f) * (x * (1.0f / (fabsf(x) + 1.0f)))), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.10000000149011612))
		tmp = copysign(Float32(Float32(x * Float32(0.5)) * Float32(x * Float32(Float32(1.0) / Float32(abs(x) + Float32(1.0))))), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(0.10000000149011612))
		tmp = sign(x) * abs(((x * single(0.5)) * (x * (single(1.0) / (abs(x) + single(1.0))))));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.100000001

   1. Initial program 31.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f32 N/A

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

      12. +-lowering-+.f32 N/A

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

      13. fabs-lowering-fabs.f32 N/A

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

      14. accelerator-lowering-log1p.f32 N/A

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

      15. fabs-lowering-fabs.f32 71.6

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

   5. Simplified 71.6%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f32 N/A

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

      3. +-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + 0}}{1 + \left|x\right|}, x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\frac{\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 0}{1 + \left|x\right|}, x\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot x} + 0}{1 + \left|x\right|}, x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} + 0}{1 + \left|x\right|}, x\right) \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x, 0\right)}}{1 + \left|x\right|}, x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, 0\right)}{1 + \left|x\right|}, x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, 0\right)}{1 + \left|x\right|}, x\right) \]

      10. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, 0\right)}{\color{blue}{1 + \left|x\right|}}, x\right) \]

      11. fabs-lowering-fabs.f32 12.9

          \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, x \cdot 0.5, 0\right)}{1 + \color{blue}{\left|x\right|}}, x\right) \]

   8. Simplified 12.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, 0\right)}{1 + \left|x\right|}}, x\right) \]
   9. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot \left(x \cdot \frac{1}{2}\right) + 0\right) \cdot \frac{1}{1 + \left|x\right|}}, x\right) \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f32 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. *-lowering-*.f32 N/A

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

      8. /-lowering-/.f32 N/A

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

      9. +-lowering-+.f32 N/A

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

      10. fabs-lowering-fabs.f32 13.1

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

   10. Applied egg-rr 13.1%

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

   if 0.100000001 < x

   1. Initial program 60.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. log-lowering-log.f32 43.2

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

   5. Simplified 43.2%

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

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot \frac{1}{\left|x\right| + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around 0

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

   1. accelerator-lowering-log1p.f32 N/A

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

   2. fabs-lowering-fabs.f32 70.1

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

5. Simplified 70.1%

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

Alternative 12: 12.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot \frac{1}{\left|x\right| + 1}\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (* (* x 0.5) (* x (/ 1.0 (+ (fabs x) 1.0)))) x))

C

float code(float x) {
	return copysignf(((x * 0.5f) * (x * (1.0f / (fabsf(x) + 1.0f)))), x);
}

Julia

function code(x)
	return copysign(Float32(Float32(x * Float32(0.5)) * Float32(x * Float32(Float32(1.0) / Float32(abs(x) + Float32(1.0))))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(((x * single(0.5)) * (x * (single(1.0) / (abs(x) + single(1.0))))));
end

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. /-lowering-/.f32 N/A

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

   12. +-lowering-+.f32 N/A

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

   13. fabs-lowering-fabs.f32 N/A

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

   14. accelerator-lowering-log1p.f32 N/A

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

   15. fabs-lowering-fabs.f32 57.2

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

5. Simplified 57.2%

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

6. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. +-rgt-identity N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + 0}}{1 + \left|x\right|}, x\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 0}{1 + \left|x\right|}, x\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot x} + 0}{1 + \left|x\right|}, x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} + 0}{1 + \left|x\right|}, x\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x, 0\right)}}{1 + \left|x\right|}, x\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, 0\right)}{1 + \left|x\right|}, x\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, 0\right)}{1 + \left|x\right|}, x\right) \]

   10. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, 0\right)}{\color{blue}{1 + \left|x\right|}}, x\right) \]

   11. fabs-lowering-fabs.f32 12.5

       \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, x \cdot 0.5, 0\right)}{1 + \color{blue}{\left|x\right|}}, x\right) \]

8. Simplified 12.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, 0\right)}{1 + \left|x\right|}}, x\right) \]
9. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot \left(x \cdot \frac{1}{2}\right) + 0\right) \cdot \frac{1}{1 + \left|x\right|}}, x\right) \]

   2. +-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. /-lowering-/.f32 N/A

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

   9. +-lowering-+.f32 N/A

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

   10. fabs-lowering-fabs.f32 12.9

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

10. Applied egg-rr 12.9%

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

11. Final simplification 12.9%

    \[\leadsto \mathsf{copysign}\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot \frac{1}{\left|x\right| + 1}\right), x\right) \]
12. Add Preprocessing

Alternative 13: 12.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\left(x \cdot 0.5\right) \cdot \frac{x}{\left|x\right| + 1}, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (* (* x 0.5) (/ x (+ (fabs x) 1.0))) x))

C

float code(float x) {
	return copysignf(((x * 0.5f) * (x / (fabsf(x) + 1.0f))), x);
}

Julia

function code(x)
	return copysign(Float32(Float32(x * Float32(0.5)) * Float32(x / Float32(abs(x) + Float32(1.0)))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(((x * single(0.5)) * (x / (abs(x) + single(1.0)))));
end

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. /-lowering-/.f32 N/A

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

   12. +-lowering-+.f32 N/A

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

   13. fabs-lowering-fabs.f32 N/A

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

   14. accelerator-lowering-log1p.f32 N/A

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

   15. fabs-lowering-fabs.f32 57.2

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

5. Simplified 57.2%

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

6. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. +-rgt-identity N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + 0}}{1 + \left|x\right|}, x\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 0}{1 + \left|x\right|}, x\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot x} + 0}{1 + \left|x\right|}, x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} + 0}{1 + \left|x\right|}, x\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x, 0\right)}}{1 + \left|x\right|}, x\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, 0\right)}{1 + \left|x\right|}, x\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, 0\right)}{1 + \left|x\right|}, x\right) \]

   10. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, 0\right)}{\color{blue}{1 + \left|x\right|}}, x\right) \]

   11. fabs-lowering-fabs.f32 12.5

       \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(x, x \cdot 0.5, 0\right)}{1 + \color{blue}{\left|x\right|}}, x\right) \]

8. Simplified 12.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, 0\right)}{1 + \left|x\right|}}, x\right) \]
9. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. +-lowering-+.f32 N/A

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

   8. fabs-lowering-fabs.f32 12.9

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

10. Applied egg-rr 12.9%

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

11. Final simplification 12.9%

    \[\leadsto \mathsf{copysign}\left(\left(x \cdot 0.5\right) \cdot \frac{x}{\left|x\right| + 1}, x\right) \]
12. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024194 
(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))