Rust f32::asinh

Percentage Accurate: 37.4% → 98.3%

Time: 10.2s

Alternatives: 14

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: 37.4% 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: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(-0.041666666666666664, \frac{x \cdot x}{t\_1} \cdot \left(3 + \frac{3}{t\_1}\right), \frac{0.5}{t\_1}\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (+ (fabs x) 1.0)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (fma
         x
         (*
          x
          (fma
           -0.041666666666666664
           (* (/ (* x x) t_1) (+ 3.0 (/ 3.0 t_1)))
           (/ 0.5 t_1)))
         (log1p (fabs x)))
        x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = fabsf(x) + 1.0f;
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf(fmaf(x, (x * fmaf(-0.041666666666666664f, (((x * x) / t_1) * (3.0f + (3.0f / t_1))), (0.5f / t_1))), log1pf(fabsf(x))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = Float32(abs(x) + Float32(1.0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(fma(x, Float32(x * fma(Float32(-0.041666666666666664), Float32(Float32(Float32(x * x) / t_1) * Float32(Float32(3.0) + Float32(Float32(3.0) / t_1))), Float32(Float32(0.5) / t_1))), log1p(abs(x))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 59.5%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.0%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f32 N/A

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

   5. Applied rewrites 99.6%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf

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

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

      7. associate-*l/ N/A

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

      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. lower-+.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. lower-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

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(-0.041666666666666664, \frac{x \cdot x}{\left|x\right| + 1} \cdot \left(3 + \frac{3}{\left|x\right| + 1}\right), \frac{0.5}{\left|x\right| + 1}\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \frac{x \cdot x}{\left|x\right| + 1}\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(t\_1, \mathsf{fma}\left(-0.125, t\_1, 0.5\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (/ (* x x) (+ (fabs x) 1.0))))
   (if (<= t_0 -1.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign (fma t_1 (fma -0.125 t_1 0.5) (log1p (fabs x))) x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = (x * x) / (fabsf(x) + 1.0f);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf(fmaf(t_1, fmaf(-0.125f, t_1, 0.5f), log1pf(fabsf(x))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + (0.5f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = Float32(Float32(x * x) / Float32(abs(x) + Float32(1.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(fma(t_1, fma(Float32(-0.125), t_1, Float32(0.5)), 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) < -1

   1. Initial program 59.5%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.0%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-fabs.f32 21.3

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

   5. Applied rewrites 21.3%

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

      1. lift-*.f32 N/A

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

      2. lift-fabs.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lower-log1p.f32 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.1%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf

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

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

      7. associate-*l/ N/A

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

      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. lower-+.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. lower-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

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\frac{x \cdot x}{\left|x\right| + 1}, \mathsf{fma}\left(-0.125, \frac{x \cdot x}{\left|x\right| + 1}, 0.5\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_0 0.05000000074505806)
       (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 <= -1.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (t_0 <= 0.05000000074505806f) {
		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(-1.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		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) < -1

   1. Initial program 59.5%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.0%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-fabs.f32 21.3

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

   5. Applied rewrites 21.3%

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

      1. lift-*.f32 N/A

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

      2. lift-fabs.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lower-log1p.f32 99.1

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

   7. Applied rewrites 99.1%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf

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

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

      7. associate-*l/ N/A

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

      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. lower-+.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. lower-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

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

   5. Applied rewrites 99.5%

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

Alternative 4: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (/ -0.5 x)) x)
     (if (<= t_0 0.05000000074505806)
       (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 <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		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(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		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) < -1

   1. Initial program 59.5%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f32 97.0

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

   8. Applied rewrites 97.0%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-fabs.f32 21.3

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

   5. Applied rewrites 21.3%

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

      1. lift-*.f32 N/A

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

      2. lift-fabs.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lower-log1p.f32 99.1

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

   7. Applied rewrites 99.1%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf

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

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

      7. associate-*l/ N/A

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

      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. lower-+.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. lower-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

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

   5. Applied rewrites 99.5%

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

Alternative 5: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (/ -0.5 x)) x)
     (if (<= t_0 0.05000000074505806)
       (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 <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		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(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		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) < -1

   1. Initial program 59.5%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f32 97.0

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

   8. Applied rewrites 97.0%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-fabs.f32 21.3

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

   5. Applied rewrites 21.3%

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

      1. lift-*.f32 N/A

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

      2. lift-fabs.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lower-log1p.f32 99.1

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

   7. Applied rewrites 99.1%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f32 N/A

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

      8. lower-fabs.f32 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 6: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{fma}\left(x, x, \mathsf{log1p}\left(-\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (/ -0.5 x)) x)
     (if (<= t_0 0.05000000074505806)
       (copysign (- (fma x x (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 <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf(-fmaf(x, x, 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(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(-fma(x, x, log1p(Float32(-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) < -1

   1. Initial program 59.5%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f32 97.0

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

   8. Applied rewrites 97.0%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f32 N/A

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

      2. lower-fabs.f32 94.1

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

   5. Applied rewrites 94.1%

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

      1. lift-fabs.f32 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f32 N/A

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

      6. lower-log.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-fabs.f32 N/A

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

      9. sqr-abs N/A

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

      10. *-rgt-identity N/A

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

      11. lower-/.f32 N/A

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

      12. lower--.f32 N/A

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

      13. metadata-eval N/A

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

      14. *-rgt-identity N/A

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

      15. lower--.f32 N/A

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

      16. lower-*.f32 19.6

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

   7. Applied rewrites 19.6%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

      4. +-commutative N/A

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

      5. lower-neg.f32 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f32 N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f32 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f32 N/A

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

      14. lower-fabs.f32 94.2

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

   10. Applied rewrites 94.2%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f32 N/A

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

      8. lower-fabs.f32 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 7: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (/ -0.5 x)) x)
     (if (<= t_0 0.05000000074505806)
       (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 <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		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(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		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) < -1

   1. Initial program 59.5%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f32 97.0

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

   8. Applied rewrites 97.0%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f32 N/A

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

      2. lower-fabs.f32 94.1

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

   5. Applied rewrites 94.1%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f32 N/A

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

      8. lower-fabs.f32 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 8: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.05000000074505806)
       (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 <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		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(-1.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		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) < -1

   1. Initial program 59.5%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. lower--.f32 N/A

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

      16. lower-fabs.f32 97.0

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

   5. Applied rewrites 97.0%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f32 N/A

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

      2. lower-fabs.f32 94.1

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

   5. Applied rewrites 94.1%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f32 N/A

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

      8. lower-fabs.f32 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 9: 21.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x) <= Float32(2.0))
		tmp = copysign(Float32(x * Float32(Float32(x * Float32(0.5)) / 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 ((sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))))) <= single(2.0))
		tmp = sign(x) * abs((x * ((x * single(0.5)) / (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 (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 2

   1. Initial program 35.6%

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

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-fabs.f32 N/A

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

      14. lower-log1p.f32 N/A

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

      15. lower-fabs.f32 67.2

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

   5. Applied rewrites 67.2%

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

   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. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f32 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-fabs.f32 13.4

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

   8. Applied rewrites 13.4%

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

      1. lift-fabs.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 13.4

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

      6. lift-*.f32 N/A

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

      7. lift-/.f32 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. lower-/.f32 13.4

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f32 13.4

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

   10. Applied rewrites 13.4%

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

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

   1. Initial program 59.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log.f32 44.2

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

   5. Applied rewrites 44.2%

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

4. Final simplification 21.8%

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

Alternative 10: 82.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.05000000074505806)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ x (fabs x))) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.05000000074505806f) {
		tmp = copysignf(log1pf(fabsf(x)), x);
	} else {
		tmp = copysignf(logf((x + fabsf(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.05000000074505806))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(x + abs(x))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0500000007

   1. Initial program 35.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f32 N/A

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

      2. lower-fabs.f32 75.8

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

   5. Applied rewrites 75.8%

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

   if 0.0500000007 < x

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f32 N/A

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

      8. lower-fabs.f32 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 11: 68.8% 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 42.3%

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

3. Taylor expanded in x around 0

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

   1. lower-log1p.f32 N/A

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

   2. lower-fabs.f32 67.0

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

5. Applied rewrites 67.0%

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

Alternative 12: 12.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-+.f32 N/A

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

   13. lower-fabs.f32 N/A

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

   14. lower-log1p.f32 N/A

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

   15. lower-fabs.f32 51.7

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

5. Applied rewrites 51.7%

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

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. associate-*l/ N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-+.f32 N/A

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

   13. lower-fabs.f32 12.8

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

8. Applied rewrites 12.8%

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

   1. lift-fabs.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 12.8

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l/ N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lower-/.f32 12.8

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

   12. lift-+.f32 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f32 12.8

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

10. Applied rewrites 12.8%

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

11. Final simplification 12.8%

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

Alternative 13: 12.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-+.f32 N/A

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

   13. lower-fabs.f32 N/A

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

   14. lower-log1p.f32 N/A

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

   15. lower-fabs.f32 51.7

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

5. Applied rewrites 51.7%

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

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. associate-*l/ N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-+.f32 N/A

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

   13. lower-fabs.f32 12.8

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

8. Applied rewrites 12.8%

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

9. Final simplification 12.8%

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

Alternative 14: 11.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in x around 0

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

   1. lower-log1p.f32 N/A

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

   2. lower-fabs.f32 67.0

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

5. Applied rewrites 67.0%

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

   1. lift-fabs.f32 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. log-rec N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-log.f32 N/A

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

   7. lift-fabs.f32 N/A

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

   8. lift-fabs.f32 N/A

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

   9. sqr-abs N/A

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

   10. *-rgt-identity N/A

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

   11. lower-/.f32 N/A

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

   12. lower--.f32 N/A

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

   13. metadata-eval N/A

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

   14. *-rgt-identity N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 23.3

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

7. Applied rewrites 23.3%

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

8. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

   4. +-commutative N/A

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

   5. lower-neg.f32 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f32 N/A

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

   9. sub-neg N/A

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

   10. mul-1-neg N/A

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

   11. lower-log1p.f32 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-fabs.f32 43.4

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

10. Applied rewrites 43.4%

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

11. Taylor expanded in x around inf

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

    1. mul-1-neg N/A

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{neg}\left({x}^{2}\right)}, x\right) \]

    2. lower-neg.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{neg}\left({x}^{2}\right)}, x\right) \]

    3. unpow2 N/A

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

    4. lower-*.f32 11.3

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

13. Applied rewrites 11.3%

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

14. Final simplification 11.3%

    \[\leadsto \mathsf{copysign}\left(x \cdot \left(-x\right), x\right) \]
15. 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 2024214 
(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))