Rust f32::asinh

Percentage Accurate: 37.8% → 99.6%

Time: 7.4s

Alternatives: 12

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: 37.8% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (+
         x
         (*
          (pow x 3.0)
          (fma
           (pow x 2.0)
           (fma (pow x 2.0) -0.044642857142857144 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 <= -0.20000000298023224f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf((x + (powf(x, 3.0f) * fmaf(powf(x, 2.0f), fmaf(powf(x, 2.0f), -0.044642857142857144f, 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(-0.20000000298023224))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * fma((x ^ Float32(2.0)), fma((x ^ Float32(2.0)), Float32(-0.044642857142857144), 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) < -0.200000003

   1. Initial program 55.2%

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

      1. +-commutative 55.2%

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

      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. flip-+ 9.4%

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

      4. hypot-1-def 9.3%

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

      5. hypot-1-def 9.3%

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

      6. add-sqr-sqrt 9.1%

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

      7. +-commutative 9.1%

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

      8. hypot-1-def 9.1%

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

      9. +-commutative 9.1%

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

      10. div-sub 9.0%

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

   4. Applied egg-rr 12.3%

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

      1. div-sub 15.5%

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

      2. remove-double-neg 15.5%

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

      3. distribute-neg-frac2 15.5%

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

      4. distribute-frac-neg 15.5%

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

      5. neg-sub0 15.5%

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

      6. associate--r- 15.5%

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

      7. neg-sub0 15.5%

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

      8. +-commutative 15.5%

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

      9. fma-undefine 15.5%

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

      10. unpow2 15.5%

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

      11. +-commutative 15.5%

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

      12. associate-+l+ 52.1%

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

      13. sub-neg 52.1%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 23.8%

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

      1. add-log-exp 23.8%

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

      2. *-un-lft-identity 23.8%

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

      3. log-prod 23.8%

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

      4. metadata-eval 23.8%

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

      5. add-log-exp 23.8%

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

      6. add-sqr-sqrt 11.7%

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

      7. fabs-sqr 11.7%

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

      8. add-sqr-sqrt 24.1%

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

   4. Applied egg-rr 24.1%

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

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot 1 + x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right)}, x\right) \]

      2. *-rgt-identity 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right), x\right) \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)}, x\right) \]

      4. unpow2 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right), x\right) \]

      5. cube-mult 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{{x}^{3}} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right), x\right) \]

      6. fmm-def 100.0%

         \[\leadsto \mathsf{copysign}\left(x + {x}^{3} \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, 0.075 + -0.044642857142857144 \cdot {x}^{2}, -0.16666666666666666\right)}, x\right) \]

      7. +-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + {x}^{3} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{-0.044642857142857144 \cdot {x}^{2} + 0.075}, -0.16666666666666666\right), x\right) \]

      8. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + {x}^{3} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{{x}^{2} \cdot -0.044642857142857144} + 0.075, -0.16666666666666666\right), x\right) \]

      9. fma-define 100.0%

         \[\leadsto \mathsf{copysign}\left(x + {x}^{3} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right)}, -0.16666666666666666\right), x\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \mathsf{copysign}\left(x + {x}^{3} \cdot \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), \color{blue}{-0.16666666666666666}\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + {x}^{3} \cdot \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right)}, x\right) \]

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

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0 56.9%

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

      1. rem-square-sqrt 56.9%

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

      2. fabs-sqr 56.9%

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

      3. metadata-eval 56.9%

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

      4. unpow2 56.9%

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

      5. hypot-undefine 99.7%

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

      6. rem-square-sqrt 99.8%

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

   5. Simplified 99.8%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (*
         x
         (+ 1.0 (* (pow x 2.0) (- (* (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 <= -0.20000000298023224f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((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(-0.20000000298023224))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((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

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(-0.20000000298023224))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (t_0 <= single(0.05000000074505806))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x ^ single(2.0)) * single(0.075)) - 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) < -0.200000003

   1. Initial program 55.2%

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

      1. +-commutative 55.2%

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

      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. flip-+ 9.4%

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

      4. hypot-1-def 9.3%

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

      5. hypot-1-def 9.3%

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

      6. add-sqr-sqrt 9.1%

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

      7. +-commutative 9.1%

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

      8. hypot-1-def 9.1%

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

      9. +-commutative 9.1%

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

      10. div-sub 9.0%

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

   4. Applied egg-rr 12.3%

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

      1. div-sub 15.5%

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

      2. remove-double-neg 15.5%

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

      3. distribute-neg-frac2 15.5%

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

      4. distribute-frac-neg 15.5%

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

      5. neg-sub0 15.5%

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

      6. associate--r- 15.5%

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

      7. neg-sub0 15.5%

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

      8. +-commutative 15.5%

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

      9. fma-undefine 15.5%

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

      10. unpow2 15.5%

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

      11. +-commutative 15.5%

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

      12. associate-+l+ 52.1%

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

      13. sub-neg 52.1%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 23.8%

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

      1. add-log-exp 23.8%

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

      2. *-un-lft-identity 23.8%

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

      3. log-prod 23.8%

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

      4. metadata-eval 23.8%

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

      5. add-log-exp 23.8%

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

      6. add-sqr-sqrt 11.7%

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

      7. fabs-sqr 11.7%

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

      8. add-sqr-sqrt 24.1%

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

   4. Applied egg-rr 24.1%

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

   5. Taylor expanded in x around 0 99.9%

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

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

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0 56.9%

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

      1. rem-square-sqrt 56.9%

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

      2. fabs-sqr 56.9%

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

      3. metadata-eval 56.9%

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

      4. unpow2 56.9%

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

      5. hypot-undefine 99.7%

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

      6. rem-square-sqrt 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.10000000149011612)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.10000000149011612f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (x <= 0.05000000074505806f) {
		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)
	tmp = Float32(0.0)
	if (x <= Float32(-0.10000000149011612))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.05000000074505806))
		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)
	tmp = single(0.0);
	if (x <= single(-0.10000000149011612))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (x <= single(0.05000000074505806))
		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 x < -0.100000001

   1. Initial program 56.2%

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

      1. +-commutative 56.2%

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

      2. hypot-1-def 99.6%

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

      3. flip-+ 11.6%

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

      4. hypot-1-def 11.6%

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

      5. hypot-1-def 11.5%

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

      6. add-sqr-sqrt 11.4%

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

      7. +-commutative 11.4%

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

      8. hypot-1-def 11.4%

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

      9. +-commutative 11.4%

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

      10. div-sub 11.3%

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

   4. Applied egg-rr 14.5%

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

      1. div-sub 17.6%

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

      2. remove-double-neg 17.6%

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

      3. distribute-neg-frac2 17.6%

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

      4. distribute-frac-neg 17.6%

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

      5. neg-sub0 17.6%

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

      6. associate--r- 17.6%

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

      7. neg-sub0 17.6%

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

      8. +-commutative 17.6%

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

      9. fma-undefine 17.6%

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

      10. unpow2 17.6%

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

      11. +-commutative 17.6%

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

      12. associate-+l+ 53.2%

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

      13. sub-neg 53.2%

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

      14. +-inverses 99.7%

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

      15. metadata-eval 99.7%

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

      16. neg-sub0 99.7%

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

      17. associate--r- 99.7%

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

      18. neg-sub0 99.7%

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

      19. +-commutative 99.7%

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

      20. sub-neg 99.7%

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

   6. Simplified 99.7%

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

   if -0.100000001 < x < 0.0500000007

   1. Initial program 22.7%

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

      1. add-log-exp 22.7%

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

      2. *-un-lft-identity 22.7%

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

      3. log-prod 22.7%

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

      4. metadata-eval 22.7%

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

      5. add-log-exp 22.7%

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

      6. add-sqr-sqrt 11.8%

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

      7. fabs-sqr 11.8%

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

      8. add-sqr-sqrt 23.0%

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

   4. Applied egg-rr 23.0%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. distribute-lft-in 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 0.0500000007 < x

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0 56.9%

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

      1. rem-square-sqrt 56.9%

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

      2. fabs-sqr 56.9%

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

      3. metadata-eval 56.9%

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

      4. unpow2 56.9%

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

      5. hypot-undefine 99.7%

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

      6. rem-square-sqrt 99.8%

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

   5. Simplified 99.8%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.10000000149011612)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.10000000149011612f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.05000000074505806f) {
		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)
	tmp = Float32(0.0)
	if (x <= Float32(-0.10000000149011612))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.05000000074505806))
		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)
	tmp = single(0.0);
	if (x <= single(-0.10000000149011612))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.05000000074505806))
		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 x < -0.100000001

   1. Initial program 56.2%

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

      1. flip-+ 11.6%

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

      2. clear-num 11.6%

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

      3. log-div 11.6%

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

      4. metadata-eval 11.6%

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

      5. +-commutative 11.6%

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

      6. hypot-1-def 11.6%

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

      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 12.6%

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

   4. Applied egg-rr 15.1%

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

      1. neg-sub0 15.1%

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

      2. div-sub 15.1%

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

      3. *-rgt-identity 15.1%

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

      4. associate-/l* 15.1%

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

      5. fma-undefine 15.1%

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

      6. unpow2 15.1%

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

      7. associate--r+ 15.1%

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

      8. +-inverses 15.1%

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

      9. metadata-eval 15.1%

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

      10. metadata-eval 15.1%

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

      11. *-commutative 15.1%

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

      12. neg-mul-1 15.1%

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

      13. neg-sub0 15.1%

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

      14. *-rgt-identity 15.1%

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

      15. associate-/l* 15.1%

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

      16. fma-undefine 15.1%

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

      17. unpow2 15.1%

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

   6. Simplified 99.6%

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

   if -0.100000001 < x < 0.0500000007

   1. Initial program 22.7%

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

      1. add-log-exp 22.7%

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

      2. *-un-lft-identity 22.7%

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

      3. log-prod 22.7%

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

      4. metadata-eval 22.7%

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

      5. add-log-exp 22.7%

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

      6. add-sqr-sqrt 11.8%

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

      7. fabs-sqr 11.8%

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

      8. add-sqr-sqrt 23.0%

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

   4. Applied egg-rr 23.0%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. distribute-lft-in 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 0.0500000007 < x

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0 56.9%

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

      1. rem-square-sqrt 56.9%

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

      2. fabs-sqr 56.9%

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

      3. metadata-eval 56.9%

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

      4. unpow2 56.9%

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

      5. hypot-undefine 99.7%

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

      6. rem-square-sqrt 99.8%

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

   5. Simplified 99.8%

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

Alternative 5: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		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)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		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)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.05000000074505806))
		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 x < -2

   1. Initial program 53.9%

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

      1. add-log-exp 10.6%

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

      2. *-un-lft-identity 10.6%

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

      3. log-prod 10.6%

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

      4. metadata-eval 10.6%

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

      5. add-log-exp 53.9%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 12.3%

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

   4. Applied egg-rr 12.3%

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

   5. Taylor expanded in x around -inf 98.2%

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

   if -2 < x < 0.0500000007

   1. Initial program 24.9%

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

      1. add-log-exp 24.9%

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

      2. *-un-lft-identity 24.9%

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

      3. log-prod 24.9%

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

      4. metadata-eval 24.9%

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

      5. add-log-exp 24.9%

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

      6. add-sqr-sqrt 11.5%

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

      7. fabs-sqr 11.5%

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

      8. add-sqr-sqrt 25.2%

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

   4. Applied egg-rr 25.2%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. distribute-lft-in 98.8%

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

      2. *-rgt-identity 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*r* 98.8%

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

      5. unpow2 98.8%

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

      6. cube-mult 98.8%

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

   7. Simplified 98.8%

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

   if 0.0500000007 < x

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0 56.9%

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

      1. rem-square-sqrt 56.9%

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

      2. fabs-sqr 56.9%

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

      3. metadata-eval 56.9%

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

      4. unpow2 56.9%

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

      5. hypot-undefine 99.7%

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

      6. rem-square-sqrt 99.8%

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

   5. Simplified 99.8%

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

Alternative 6: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\frac{0.5}{x}}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.4000000059604645)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (/ 1.0 (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((1.0f / (0.5f / x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(1.0) / Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((single(1.0) / (single(0.5) / x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 53.9%

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

      1. add-log-exp 10.6%

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

      2. *-un-lft-identity 10.6%

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

      3. log-prod 10.6%

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

      4. metadata-eval 10.6%

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

      5. add-log-exp 53.9%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 12.3%

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

   4. Applied egg-rr 12.3%

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

   5. Taylor expanded in x around -inf 98.2%

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

   if -2 < x < 0.400000006

   1. Initial program 26.0%

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

      1. add-log-exp 26.0%

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

      2. *-un-lft-identity 26.0%

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

      3. log-prod 26.0%

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

      4. metadata-eval 26.0%

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

      5. add-log-exp 26.0%

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

      6. add-sqr-sqrt 12.8%

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

      7. fabs-sqr 12.8%

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

      8. add-sqr-sqrt 26.3%

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

   4. Applied egg-rr 26.3%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. distribute-lft-in 98.3%

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

      2. *-rgt-identity 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*r* 98.3%

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

      5. unpow2 98.3%

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

      6. cube-mult 98.3%

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

   7. Simplified 98.3%

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

   if 0.400000006 < x

   1. Initial program 55.6%

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

      1. +-commutative 55.6%

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

      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. flip-+ 14.0%

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

      4. hypot-1-def 13.8%

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

      5. hypot-1-def 13.7%

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

      6. add-sqr-sqrt 13.7%

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

      7. +-commutative 13.7%

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

      8. hypot-1-def 13.6%

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

      9. +-commutative 13.6%

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

      10. div-sub 14.1%

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

   4. Applied egg-rr 14.0%

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

      1. div-sub 13.7%

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

      2. remove-double-neg 13.7%

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

      3. distribute-neg-frac2 13.7%

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

      4. distribute-frac-neg 13.7%

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

      5. neg-sub0 13.7%

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

      6. associate--r- 13.7%

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

      7. neg-sub0 13.7%

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

      8. +-commutative 13.7%

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

      9. fma-undefine 13.7%

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

      10. unpow2 13.7%

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

      11. +-commutative 13.7%

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

      12. associate-+l+ 13.7%

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

      13. sub-neg 13.7%

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

      14. +-inverses 13.7%

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

      15. metadata-eval 13.7%

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

      16. neg-sub0 18.9%

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

      17. associate--r- 18.9%

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

      18. neg-sub0 18.9%

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

      19. +-commutative 18.9%

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

      20. sub-neg 18.9%

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

   6. Simplified 18.9%

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

   7. Taylor expanded in x around inf 95.0%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\frac{0.5}{x}}}\right), x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\frac{0.5}{x}}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.4000000059604645)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log (/ 1.0 (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(logf((1.0f / (0.5f / x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float32(Float32(1.0) / Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * single(-0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((single(1.0) / (single(0.5) / x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 53.9%

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

      1. add-log-exp 10.6%

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

      2. *-un-lft-identity 10.6%

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

      3. log-prod 10.6%

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

      4. metadata-eval 10.6%

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

      5. add-log-exp 53.9%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 12.3%

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

   4. Applied egg-rr 12.3%

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

   5. Taylor expanded in x around -inf 98.2%

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

   if -2 < x < 0.400000006

   1. Initial program 26.0%

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

      1. add-log-exp 26.0%

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

      2. *-un-lft-identity 26.0%

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

      3. log-prod 26.0%

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

      4. metadata-eval 26.0%

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

      5. add-log-exp 26.0%

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

      6. add-sqr-sqrt 12.8%

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

      7. fabs-sqr 12.8%

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

      8. add-sqr-sqrt 26.3%

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

   4. Applied egg-rr 26.3%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. unpow2 98.2%

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

   9. Applied egg-rr 98.2%

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

   if 0.400000006 < x

   1. Initial program 55.6%

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

      1. +-commutative 55.6%

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

      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. flip-+ 14.0%

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

      4. hypot-1-def 13.8%

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

      5. hypot-1-def 13.7%

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

      6. add-sqr-sqrt 13.7%

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

      7. +-commutative 13.7%

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

      8. hypot-1-def 13.6%

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

      9. +-commutative 13.6%

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

      10. div-sub 14.1%

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

   4. Applied egg-rr 14.0%

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

      1. div-sub 13.7%

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

      2. remove-double-neg 13.7%

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

      3. distribute-neg-frac2 13.7%

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

      4. distribute-frac-neg 13.7%

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

      5. neg-sub0 13.7%

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

      6. associate--r- 13.7%

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

      7. neg-sub0 13.7%

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

      8. +-commutative 13.7%

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

      9. fma-undefine 13.7%

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

      10. unpow2 13.7%

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

      11. +-commutative 13.7%

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

      12. associate-+l+ 13.7%

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

      13. sub-neg 13.7%

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

      14. +-inverses 13.7%

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

      15. metadata-eval 13.7%

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

      16. neg-sub0 18.9%

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

      17. associate--r- 18.9%

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

      18. neg-sub0 18.9%

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

      19. +-commutative 18.9%

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

      20. sub-neg 18.9%

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

   6. Simplified 18.9%

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

   7. Taylor expanded in x around inf 95.0%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\frac{0.5}{x}}}\right), x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.4000000059604645)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * 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(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * 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 < -2

   1. Initial program 53.9%

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

      1. add-log-exp 10.6%

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

      2. *-un-lft-identity 10.6%

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

      3. log-prod 10.6%

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

      4. metadata-eval 10.6%

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

      5. add-log-exp 53.9%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 12.3%

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

   4. Applied egg-rr 12.3%

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

   5. Taylor expanded in x around -inf 98.2%

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

   if -2 < x < 0.400000006

   1. Initial program 26.0%

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

      1. add-log-exp 26.0%

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

      2. *-un-lft-identity 26.0%

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

      3. log-prod 26.0%

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

      4. metadata-eval 26.0%

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

      5. add-log-exp 26.0%

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

      6. add-sqr-sqrt 12.8%

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

      7. fabs-sqr 12.8%

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

      8. add-sqr-sqrt 26.3%

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

   4. Applied egg-rr 26.3%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. unpow2 98.2%

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

   9. Applied egg-rr 98.2%

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

   if 0.400000006 < x

   1. Initial program 55.6%

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

   3. Taylor expanded in x around inf 95.0%

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

      1. +-commutative 95.0%

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

      2. rem-square-sqrt 95.0%

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

      3. fabs-sqr 95.0%

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

      4. rem-square-sqrt 95.0%

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

      5. *-inverses 95.0%

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

      6. metadata-eval 95.0%

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

   5. Simplified 95.0%

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

Alternative 9: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- x)) x)
   (if (<= x 0.4000000059604645)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * 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(-2.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * 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 < -2

   1. Initial program 53.9%

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

   3. Taylor expanded in x around -inf 44.2%

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

      1. neg-mul-1 44.2%

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

   5. Simplified 44.2%

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

   if -2 < x < 0.400000006

   1. Initial program 26.0%

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

      1. add-log-exp 26.0%

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

      2. *-un-lft-identity 26.0%

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

      3. log-prod 26.0%

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

      4. metadata-eval 26.0%

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

      5. add-log-exp 26.0%

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

      6. add-sqr-sqrt 12.8%

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

      7. fabs-sqr 12.8%

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

      8. add-sqr-sqrt 26.3%

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

   4. Applied egg-rr 26.3%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. unpow2 98.2%

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

   9. Applied egg-rr 98.2%

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

   if 0.400000006 < x

   1. Initial program 55.6%

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

   3. Taylor expanded in x around inf 95.0%

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

      1. +-commutative 95.0%

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

      2. rem-square-sqrt 95.0%

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

      3. fabs-sqr 95.0%

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

      4. rem-square-sqrt 95.0%

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

      5. *-inverses 95.0%

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

      6. metadata-eval 95.0%

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

   5. Simplified 95.0%

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

Alternative 10: 71.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 3.4000000953674316:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- x)) x)
   (if (<= x 3.4000000953674316)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log1p x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 3.4000000953674316f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(3.4000000953674316))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 53.9%

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

   3. Taylor expanded in x around -inf 44.2%

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

      1. neg-mul-1 44.2%

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

   5. Simplified 44.2%

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

   if -2 < x < 3.4000001

   1. Initial program 27.1%

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

      1. add-log-exp 27.1%

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

      2. *-un-lft-identity 27.1%

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

      3. log-prod 27.1%

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

      4. metadata-eval 27.1%

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

      5. add-log-exp 27.1%

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

      6. add-sqr-sqrt 14.1%

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

      7. fabs-sqr 14.1%

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

      8. add-sqr-sqrt 27.4%

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

   4. Applied egg-rr 27.4%

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

   5. Taylor expanded in x around 0 97.2%

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

      1. *-commutative 97.2%

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

   7. Simplified 97.2%

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

      1. unpow2 97.2%

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

   9. Applied egg-rr 97.2%

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

   if 3.4000001 < x

   1. Initial program 54.1%

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

   3. Taylor expanded in x around 0 43.9%

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

      1. log1p-define 43.9%

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

      2. rem-square-sqrt 43.9%

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

      3. fabs-sqr 43.9%

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

      4. rem-square-sqrt 43.9%

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

   5. Simplified 43.9%

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

Alternative 11: 62.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 2.0) (copysign x x) (copysign (log1p x) x)))

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 35.8%

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

      1. add-log-exp 21.2%

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

      2. *-un-lft-identity 21.2%

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

      3. log-prod 21.2%

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

      4. metadata-eval 21.2%

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

      5. add-log-exp 35.8%

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

      6. add-sqr-sqrt 8.9%

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

      7. fabs-sqr 8.9%

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

      8. add-sqr-sqrt 21.9%

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

   4. Applied egg-rr 21.9%

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

   5. Taylor expanded in x around 0 67.4%

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

   if 2 < x

   1. Initial program 54.8%

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

   3. Taylor expanded in x around 0 43.6%

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

      1. log1p-define 43.6%

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

      2. rem-square-sqrt 43.6%

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

      3. fabs-sqr 43.6%

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

      4. rem-square-sqrt 43.6%

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

   5. Simplified 43.6%

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

Alternative 12: 53.8% 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 40.1%

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

   1. add-log-exp 19.6%

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

   2. *-un-lft-identity 19.6%

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

   3. log-prod 19.6%

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

   4. metadata-eval 19.6%

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

   5. add-log-exp 40.1%

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

   6. add-sqr-sqrt 19.3%

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

   7. fabs-sqr 19.3%

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

   8. add-sqr-sqrt 29.4%

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

4. Applied egg-rr 29.4%

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

5. Taylor expanded in x around 0 54.9%

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

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024172 
(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))