Rust f32::asinh

Percentage Accurate: 37.2% → 99.4%

Time: 6.6s

Alternatives: 15

Speedup: 4.0×

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: 37.2% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot \mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.4000000059604645)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.019999999552965164)
       (copysign
        (+ x (* (pow x 3.0) (fma (pow x 2.0) 0.075 -0.16666666666666666)))
        x)
       (copysign (log (+ x (hypot 1.0 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 <= -0.4000000059604645f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.019999999552965164f) {
		tmp = copysignf((x + (powf(x, 3.0f) * fmaf(powf(x, 2.0f), 0.075f, -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, 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(-0.4000000059604645))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.019999999552965164))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * fma((x ^ Float32(2.0)), Float32(0.075), Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   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. Step-by-step derivation

      1. flip-+ 6.5%

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

      2. clear-num 6.5%

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

      3. log-div 6.5%

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

      4. metadata-eval 6.5%

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

      5. +-commutative 6.5%

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

      6. hypot-1-def 6.5%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 6.4%

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

   4. Applied egg-rr 10.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 10.1%

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

      2. div-sub 10.1%

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

      3. *-rgt-identity 10.1%

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

      4. associate-/l* 10.1%

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

      5. *-rgt-identity 10.1%

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

      6. associate-/l* 10.1%

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

      7. fma-undefine 9.5%

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

      8. unpow2 9.5%

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

      9. associate--r+ 10.2%

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

      10. +-inverses 10.2%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \frac{1}{\color{blue}{0} - 1} - \mathsf{hypot}\left(1, x\right) \cdot \frac{1}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. metadata-eval 10.2%

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

      12. metadata-eval 10.2%

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

      13. *-commutative 10.2%

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

      14. neg-mul-1 10.2%

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

      15. fma-undefine 9.5%

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

      16. unpow2 9.5%

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

      17. associate--r+ 57.1%

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

   6. Simplified 98.5%

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

   if -0.400000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0199999996

   1. Initial program 18.5%

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

      1. add-sqr-sqrt 17.4%

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

      2. pow2 17.4%

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

      3. log-pow 17.7%

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

      4. +-commutative 17.7%

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

      5. hypot-1-def 17.7%

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

      6. +-commutative 17.7%

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

      7. add-sqr-sqrt 9.1%

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

      8. fabs-sqr 9.1%

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

      9. add-sqr-sqrt 17.9%

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

   4. Applied egg-rr 17.9%

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

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot 1 + x \cdot \left({x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)}, x\right) \]

      2. *-rgt-identity 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + x \cdot \left({x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right), x\right) \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)}, x\right) \]

      4. unpow2 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right), x\right) \]

      5. cube-mult 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{{x}^{3}} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right), x\right) \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + {x}^{3} \cdot \left(\color{blue}{{x}^{2} \cdot 0.075} - 0.16666666666666666\right), x\right) \]

      7. fmm-def 100.0%

         \[\leadsto \mathsf{copysign}\left(x + {x}^{3} \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right)}, x\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \mathsf{copysign}\left(x + {x}^{3} \cdot \mathsf{fma}\left({x}^{2}, 0.075, \color{blue}{-0.16666666666666666}\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + {x}^{3} \cdot \mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right)}, x\right) \]

   if 0.0199999996 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.6%

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

      1. +-commutative 56.6%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 56.6%

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

      1. rem-square-sqrt 56.6%

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

      2. fabs-sqr 56.6%

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

      3. metadata-eval 56.6%

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

      4. unpow2 56.6%

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

      5. hypot-undefine 99.9%

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

      6. rem-square-sqrt 99.9%

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

   7. Simplified 99.9%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.10000000149011612)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.019999999552965164)
       (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
       (copysign (log (+ x (hypot 1.0 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 <= -0.10000000149011612f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.019999999552965164f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, 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(-0.10000000149011612))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.019999999552965164))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.10000000149011612))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.019999999552965164))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -0.100000001

   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. Step-by-step derivation

      1. flip-+ 7.9%

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

      2. clear-num 7.9%

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

      3. log-div 7.8%

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

      4. metadata-eval 7.8%

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

      5. +-commutative 7.8%

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

      6. hypot-1-def 7.8%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 7.8%

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

   4. Applied egg-rr 11.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 11.4%

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

      2. div-sub 11.4%

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

      3. *-rgt-identity 11.4%

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

      4. associate-/l* 11.4%

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

      5. *-rgt-identity 11.4%

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

      6. associate-/l* 11.4%

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

      7. fma-undefine 10.9%

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

      8. unpow2 10.9%

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

      9. associate--r+ 11.5%

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

      10. +-inverses 11.5%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \frac{1}{\color{blue}{0} - 1} - \mathsf{hypot}\left(1, x\right) \cdot \frac{1}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      11. metadata-eval 11.5%

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

      12. metadata-eval 11.5%

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

      13. *-commutative 11.5%

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

      14. neg-mul-1 11.5%

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

      15. fma-undefine 10.9%

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

      16. unpow2 10.9%

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

      17. associate--r+ 57.7%

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

   6. Simplified 98.5%

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

   if -0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0199999996

   1. Initial program 17.9%

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

      1. add-sqr-sqrt 16.9%

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

      2. pow2 16.9%

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

      3. log-pow 17.1%

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

      4. +-commutative 17.1%

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

      5. hypot-1-def 17.1%

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

      6. +-commutative 17.1%

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

      7. add-sqr-sqrt 9.2%

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

      8. fabs-sqr 9.2%

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

      9. add-sqr-sqrt 17.3%

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

   4. Applied egg-rr 17.3%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot 1 + x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]

      2. *-rgt-identity 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right), x\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

      4. associate-*r* 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.16666666666666666}, x\right) \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.16666666666666666, x\right) \]

      6. cube-mult 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{{x}^{3}} \cdot -0.16666666666666666, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + {x}^{3} \cdot -0.16666666666666666}, x\right) \]

   if 0.0199999996 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.6%

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

      1. +-commutative 56.6%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 56.6%

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

      1. rem-square-sqrt 56.6%

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

      2. fabs-sqr 56.6%

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

      3. metadata-eval 56.6%

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

      4. unpow2 56.6%

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

      5. hypot-undefine 99.9%

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

      6. rem-square-sqrt 99.9%

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

   7. Simplified 99.9%

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

Alternative 3: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.019999999552965164)
     (copysign (* x (+ 1.0 (* (pow x 2.0) -0.16666666666666666))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.019999999552965164f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.019999999552965164))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.019999999552965164))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * single(-0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 59.3%

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

   3. Taylor expanded in x around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. distribute-rgt-neg-in 98.2%

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

      3. mul-1-neg 98.2%

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

      4. unsub-neg 98.2%

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

      5. /-rgt-identity 98.2%

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

      6. /-rgt-identity 98.2%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 10.3%

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

      10. associate-*r/ 10.3%

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

      11. metadata-eval 10.3%

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

   5. Simplified 10.3%

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

   6. Taylor expanded in x around 0 98.9%

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

   if -5 < x < 0.0199999996

   1. Initial program 19.2%

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

      1. add-sqr-sqrt 18.0%

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

      2. pow2 18.0%

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

      3. log-pow 18.3%

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

      4. +-commutative 18.3%

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

      5. hypot-1-def 18.3%

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

      6. +-commutative 18.3%

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

      7. add-sqr-sqrt 9.0%

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

      8. fabs-sqr 9.0%

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

      9. add-sqr-sqrt 18.5%

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

   4. Applied egg-rr 18.5%

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

   5. Taylor expanded in x around 0 99.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   6. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \mathsf{copysign}\left(x \cdot \left(1 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right), x\right) \]

   7. Simplified 99.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

   if 0.0199999996 < x

   1. Initial program 56.6%

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

      1. +-commutative 56.6%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 56.6%

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

      1. rem-square-sqrt 56.6%

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

      2. fabs-sqr 56.6%

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

      3. metadata-eval 56.6%

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

      4. unpow2 56.6%

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

      5. hypot-undefine 99.9%

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

      6. rem-square-sqrt 99.9%

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

   7. Simplified 99.9%

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

Alternative 4: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (* x (+ 1.0 (* (pow x 2.0) -0.16666666666666666))) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * single(-0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 59.3%

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

   3. Taylor expanded in x around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. distribute-rgt-neg-in 98.2%

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

      3. mul-1-neg 98.2%

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

      4. unsub-neg 98.2%

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

      5. /-rgt-identity 98.2%

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

      6. /-rgt-identity 98.2%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 10.3%

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

      10. associate-*r/ 10.3%

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

      11. metadata-eval 10.3%

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

   5. Simplified 10.3%

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

   6. Taylor expanded in x around 0 98.9%

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

   if -5 < x < 1

   1. Initial program 21.6%

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

      1. add-sqr-sqrt 20.5%

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

      2. pow2 20.5%

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

      3. log-pow 20.7%

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

      4. +-commutative 20.7%

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

      5. hypot-1-def 20.7%

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

      6. +-commutative 20.7%

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

      7. add-sqr-sqrt 11.7%

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

      8. fabs-sqr 11.7%

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

      9. add-sqr-sqrt 20.9%

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

   4. Applied egg-rr 20.9%

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

   5. Taylor expanded in x around 0 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   6. Step-by-step derivation

      1. *-commutative 97.9%

         \[\leadsto \mathsf{copysign}\left(x \cdot \left(1 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right), x\right) \]

   7. Simplified 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

   if 1 < x

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf 97.2%

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

      1. +-commutative 97.2%

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

      2. rem-square-sqrt 97.2%

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

      3. fabs-sqr 97.2%

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

      4. rem-square-sqrt 97.2%

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

      5. *-inverses 97.2%

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

      6. metadata-eval 97.2%

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

   5. Simplified 97.2%

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

Alternative 5: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 59.3%

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

   3. Taylor expanded in x around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. distribute-rgt-neg-in 98.2%

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

      3. mul-1-neg 98.2%

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

      4. unsub-neg 98.2%

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

      5. /-rgt-identity 98.2%

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

      6. /-rgt-identity 98.2%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 10.3%

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

      10. associate-*r/ 10.3%

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

      11. metadata-eval 10.3%

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

   5. Simplified 10.3%

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

   6. Taylor expanded in x around 0 98.9%

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

   if -5 < x < 1

   1. Initial program 21.6%

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

      1. add-sqr-sqrt 20.5%

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

      2. pow2 20.5%

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

      3. log-pow 20.7%

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

      4. +-commutative 20.7%

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

      5. hypot-1-def 20.7%

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

      6. +-commutative 20.7%

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

      7. add-sqr-sqrt 11.7%

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

      8. fabs-sqr 11.7%

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

      9. add-sqr-sqrt 20.9%

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

   4. Applied egg-rr 20.9%

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

   5. Taylor expanded in x around 0 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-lft-in 97.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot 1 + x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]

      2. *-rgt-identity 97.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + x \cdot \left(-0.16666666666666666 \cdot {x}^{2}\right), x\right) \]

      3. *-commutative 97.9%

         \[\leadsto \mathsf{copysign}\left(x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]

      4. associate-*r* 97.9%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.16666666666666666}, x\right) \]

      5. unpow2 97.9%

         \[\leadsto \mathsf{copysign}\left(x + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.16666666666666666, x\right) \]

      6. cube-mult 97.9%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{{x}^{3}} \cdot -0.16666666666666666, x\right) \]

   7. Simplified 97.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + {x}^{3} \cdot -0.16666666666666666}, x\right) \]

   if 1 < x

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf 97.2%

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

      1. +-commutative 97.2%

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

      2. rem-square-sqrt 97.2%

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

      3. fabs-sqr 97.2%

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

      4. rem-square-sqrt 97.2%

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

      5. *-inverses 97.2%

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

      6. metadata-eval 97.2%

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

   5. Simplified 97.2%

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

Alternative 6: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 59.3%

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

   3. Taylor expanded in x around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. distribute-rgt-neg-in 98.2%

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

      3. mul-1-neg 98.2%

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

      4. unsub-neg 98.2%

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

      5. /-rgt-identity 98.2%

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

      6. /-rgt-identity 98.2%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 10.3%

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

      10. associate-*r/ 10.3%

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

      11. metadata-eval 10.3%

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

   5. Simplified 10.3%

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

   6. Taylor expanded in x around 0 98.9%

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

   if -5 < x < 1

   1. Initial program 21.6%

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

   3. Taylor expanded in x around 0 18.2%

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

      1. log1p-define 94.0%

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

      2. rem-square-sqrt 47.8%

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

      3. fabs-sqr 47.8%

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

      4. rem-square-sqrt 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in x around 0 97.0%

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

   if 1 < x

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf 97.2%

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

      1. +-commutative 97.2%

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

      2. rem-square-sqrt 97.2%

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

      3. fabs-sqr 97.2%

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

      4. rem-square-sqrt 97.2%

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

      5. *-inverses 97.2%

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

      6. metadata-eval 97.2%

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

   5. Simplified 97.2%

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

Alternative 7: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (* x -2.0)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((x * -2.0f)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(x * Float32(-2.0))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((x * single(-2.0))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 59.3%

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

   3. Taylor expanded in x around -inf 97.5%

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

      1. neg-mul-1 97.5%

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

   5. Simplified 97.5%

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

      1. +-commutative 97.5%

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

      2. neg-sub0 97.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(0 - x\right)} + \left|x\right|\right), x\right) \]

      3. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(0 - x\right) + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

      4. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(0 - x\right) + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]

      5. add-sqr-sqrt 6.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(0 - x\right) + \color{blue}{x}\right), x\right) \]

      6. associate-+l- 6.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \left(x - x\right)\right)}, x\right) \]

      7. sub-neg 6.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \color{blue}{\left(x + \left(-x\right)\right)}\right), x\right) \]

      8. add-sqr-sqrt 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \left(x + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)\right), x\right) \]

      9. sqrt-unprod 3.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \left(x + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)\right), x\right) \]

      10. sqr-neg 3.6%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \left(x + \sqrt{\color{blue}{x \cdot x}}\right)\right), x\right) \]

      11. sqrt-unprod -0.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \left(x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      12. add-sqr-sqrt 97.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \left(x + \color{blue}{x}\right)\right), x\right) \]

   7. Applied egg-rr 97.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \left(x + x\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. neg-sub0 97.5%

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

      2. count-2 97.5%

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

      3. *-commutative 97.5%

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

      4. distribute-rgt-neg-in 97.5%

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

      5. metadata-eval 97.5%

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

   9. Simplified 97.5%

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

   if -5 < x < 1

   1. Initial program 21.6%

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

   3. Taylor expanded in x around 0 18.2%

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

      1. log1p-define 94.0%

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

      2. rem-square-sqrt 47.8%

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

      3. fabs-sqr 47.8%

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

      4. rem-square-sqrt 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in x around 0 97.0%

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

   if 1 < x

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf 97.2%

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

      1. +-commutative 97.2%

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

      2. rem-square-sqrt 97.2%

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

      3. fabs-sqr 97.2%

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

      4. rem-square-sqrt 97.2%

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

      5. *-inverses 97.2%

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

      6. metadata-eval 97.2%

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

   5. Simplified 97.2%

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

Alternative 8: 82.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0) (copysign (log (* x -2.0)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((x * -2.0f)), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(x * Float32(-2.0))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -5

   1. Initial program 59.3%

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

   3. Taylor expanded in x around -inf 97.5%

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

      1. neg-mul-1 97.5%

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

   5. Simplified 97.5%

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

      1. +-commutative 97.5%

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

      2. neg-sub0 97.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(0 - x\right)} + \left|x\right|\right), x\right) \]

      3. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(0 - x\right) + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

      4. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(0 - x\right) + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]

      5. add-sqr-sqrt 6.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(0 - x\right) + \color{blue}{x}\right), x\right) \]

      6. associate-+l- 6.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \left(x - x\right)\right)}, x\right) \]

      7. sub-neg 6.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \color{blue}{\left(x + \left(-x\right)\right)}\right), x\right) \]

      8. add-sqr-sqrt 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \left(x + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)\right), x\right) \]

      9. sqrt-unprod 3.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \left(x + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)\right), x\right) \]

      10. sqr-neg 3.6%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \left(x + \sqrt{\color{blue}{x \cdot x}}\right)\right), x\right) \]

      11. sqrt-unprod -0.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \left(x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      12. add-sqr-sqrt 97.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \left(x + \color{blue}{x}\right)\right), x\right) \]

   7. Applied egg-rr 97.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \left(x + x\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. neg-sub0 97.5%

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

      2. count-2 97.5%

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

      3. *-commutative 97.5%

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

      4. distribute-rgt-neg-in 97.5%

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

      5. metadata-eval 97.5%

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

   9. Simplified 97.5%

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

   if -5 < x

   1. Initial program 32.1%

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

   3. Taylor expanded in x around 0 26.6%

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

      1. log1p-define 77.8%

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

      2. rem-square-sqrt 46.6%

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

      3. fabs-sqr 46.6%

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

      4. rem-square-sqrt 77.8%

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

   5. Simplified 77.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;{\left(\mathsf{copysign}\left(-2, -2\right)\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0) (pow (copysign -2.0 -2.0) 3.0) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = powf(copysignf(-2.0f, -2.0f), 3.0f);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(Float32(-2.0), Float32(-2.0)) ^ Float32(3.0);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -5

   1. Initial program 59.3%

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

   3. Taylor expanded in x around 0 44.1%

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

      1. log1p-define 44.1%

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

      2. rem-square-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. rem-square-sqrt -0.0%

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

   5. Simplified -0.0%

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

   7. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{{\left(\mathsf{copysign}\left(-2, -2\right)\right)}^{3}} \]

   if -5 < x

   1. Initial program 32.1%

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

   3. Taylor expanded in x around 0 26.6%

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

      1. log1p-define 77.8%

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

      2. rem-square-sqrt 46.6%

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

      3. fabs-sqr 46.6%

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

      4. rem-square-sqrt 77.8%

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

   5. Simplified 77.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0) (copysign 4.0 x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(4.0f, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(Float32(4.0), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -5

   1. Initial program 59.3%

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

   3. Taylor expanded in x around 0 44.1%

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

      1. log1p-define 44.1%

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

      2. rem-square-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. rem-square-sqrt -0.0%

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

   5. Simplified -0.0%

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

   7. Applied egg-rr 23.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{4}, x\right) \]

   if -5 < x

   1. Initial program 32.1%

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

   3. Taylor expanded in x around 0 26.6%

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

      1. log1p-define 77.8%

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

      2. rem-square-sqrt 46.6%

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

      3. fabs-sqr 46.6%

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

      4. rem-square-sqrt 77.8%

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

   5. Simplified 77.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 4.0) (copysign x x) (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 4.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(4.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(4.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 34.0%

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

   3. Taylor expanded in x around 0 26.6%

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

      1. log1p-define 77.8%

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

      2. rem-square-sqrt 32.4%

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

      3. fabs-sqr 32.4%

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

      4. rem-square-sqrt 63.6%

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

   5. Simplified 63.6%

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

   6. Taylor expanded in x around 0 69.3%

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

   if 4 < x

   1. Initial program 53.2%

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

   3. Taylor expanded in x around inf 44.2%

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

      1. mul-1-neg 44.2%

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

      2. log-rec 44.2%

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

      3. remove-double-neg 44.2%

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

   5. Simplified 44.2%

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

Alternative 12: 55.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in x around 0 30.9%

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

   1. log1p-define 69.6%

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

   2. rem-square-sqrt 35.3%

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

   3. fabs-sqr 35.3%

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

   4. rem-square-sqrt 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around 0 55.2%

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

Alternative 13: 16.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(4, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign 4.0 x))

C

float code(float x) {
	return copysignf(4.0f, x);
}

Julia

function code(x)
	return copysign(Float32(4.0), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(4.0));
end

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in x around 0 30.9%

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

   1. log1p-define 69.6%

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

   2. rem-square-sqrt 35.3%

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

   3. fabs-sqr 35.3%

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

   4. rem-square-sqrt 58.9%

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

5. Simplified 58.9%

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

7. Applied egg-rr 16.9%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{4}, x\right) \]
8. Add Preprocessing

Alternative 14: 16.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(2, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign 2.0 x))

C

float code(float x) {
	return copysignf(2.0f, x);
}

Julia

function code(x)
	return copysign(Float32(2.0), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(2.0));
end

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in x around 0 30.9%

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

   1. log1p-define 69.6%

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

   2. rem-square-sqrt 35.3%

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

   3. fabs-sqr 35.3%

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

   4. rem-square-sqrt 58.9%

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

5. Simplified 58.9%

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

7. Applied egg-rr 16.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{2}, x\right) \]
8. Add Preprocessing

Alternative 15: 16.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(1.5, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign 1.5 x))

C

float code(float x) {
	return copysignf(1.5f, x);
}

Julia

function code(x)
	return copysign(Float32(1.5), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(1.5));
end

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in x around 0 30.9%

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

   1. log1p-define 69.6%

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

   2. rem-square-sqrt 35.3%

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

   3. fabs-sqr 35.3%

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

   4. rem-square-sqrt 58.9%

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

5. Simplified 58.9%

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

7. Applied egg-rr 16.0%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{1.5}, x\right) \]
8. Add Preprocessing

Developer Target 1: 99.5% 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 2024131 
(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))