Rust f32::asinh

Percentage Accurate: 38.3% → 98.1%

Time: 8.6s

Alternatives: 11

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: 38.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

C

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

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
end

Alternative 1: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (/ -0.5 x)) x)
     (if (<= t_0 1.0)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (pow x 2.0)
           (-
            (* (pow x 2.0) (+ 0.075 (* (pow x 2.0) -0.044642857142857144)))
            0.16666666666666666))))
        x)
       (copysign (log (/ 0.5 x)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (t_0 <= 1.0f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((powf(x, 2.0f) * (0.075f + (powf(x, 2.0f) * -0.044642857142857144f))) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (t_0 <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(Float32(0.075) + Float32((x ^ Float32(2.0)) * Float32(-0.044642857142857144)))) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / 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(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (t_0 <= single(1.0))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x ^ single(2.0)) * (single(0.075) + ((x ^ single(2.0)) * single(-0.044642857142857144)))) - single(0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((single(0.5) / 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) < -5

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. sub-neg 98.7%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 8.8%

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

      10. associate-*r/ 8.8%

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

      11. metadata-eval 8.8%

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

      12. distribute-neg-frac 8.8%

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

      13. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 1

   1. Initial program 26.8%

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

      1. +-commutative 26.8%

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

      2. hypot-1-def 26.8%

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

      3. flip-+ 26.5%

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

      4. hypot-1-def 26.5%

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

      5. hypot-1-def 26.4%

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

      6. add-sqr-sqrt 26.6%

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

      7. +-commutative 26.6%

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

      8. hypot-1-def 26.6%

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

      9. +-commutative 26.6%

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

      10. div-sub 26.6%

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

   4. Applied egg-rr 26.5%

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

      1. div-sub 26.5%

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

      2. remove-double-neg 26.5%

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

      3. distribute-frac-neg2 26.5%

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

      4. neg-sub0 26.5%

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

      5. associate--r- 26.5%

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

      6. neg-sub0 26.5%

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

      7. +-commutative 26.5%

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

      8. sub-neg 26.5%

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

      9. distribute-frac-neg 26.5%

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

      10. sub-neg 26.5%

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

      11. +-commutative 26.5%

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

      12. distribute-neg-in 26.5%

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

      13. remove-double-neg 26.5%

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

      14. fma-undefine 26.5%

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

      15. unpow2 26.5%

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

      16. +-commutative 26.5%

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

      17. associate-+l+ 26.6%

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

      18. sub-neg 26.6%

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

      19. +-inverses 26.6%

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

      20. metadata-eval 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in x around 0 99.4%

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

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

   1. Initial program 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 98.5%

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

      3. flip-+ 3.1%

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

      4. hypot-1-def 3.1%

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

      5. hypot-1-def 3.1%

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

      6. add-sqr-sqrt 2.9%

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

      7. +-commutative 2.9%

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

      8. hypot-1-def 2.9%

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

      9. +-commutative 2.9%

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

      10. div-sub 2.9%

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

   4. Applied egg-rr 2.9%

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

      1. div-sub 2.9%

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

      2. remove-double-neg 2.9%

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

      3. distribute-frac-neg2 2.9%

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

      4. neg-sub0 2.9%

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

      5. associate--r- 2.9%

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

      6. neg-sub0 2.9%

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

      7. +-commutative 2.9%

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

      8. sub-neg 2.9%

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

      9. distribute-frac-neg 2.9%

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

      10. sub-neg 2.9%

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

      11. +-commutative 2.9%

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

      12. distribute-neg-in 2.9%

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

      13. remove-double-neg 2.9%

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

      14. fma-undefine 2.9%

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

      15. unpow2 2.9%

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

      16. +-commutative 2.9%

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

      17. associate-+l+ 5.6%

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

      18. sub-neg 5.6%

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

      19. +-inverses 8.9%

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

      20. metadata-eval 8.9%

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

   6. Simplified 8.9%

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

      1. inv-pow 8.9%

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

      2. sub-neg 8.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 51.8%

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

      5. sqr-neg 51.8%

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

      6. sqrt-unprod 98.5%

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

      7. add-sqr-sqrt 98.5%

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

      8. +-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

      1. unpow-1 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (/ -0.5 x)) x)
     (if (<= t_0 1.0)
       (copysign
        (*
         x
         (+ 1.0 (* (pow x 2.0) (- (* (pow x 2.0) 0.075) 0.16666666666666666))))
        x)
       (copysign (log (/ 0.5 x)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (t_0 <= 1.0f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((powf(x, 2.0f) * 0.075f) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (t_0 <= Float32(1.0))
		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(Float32(0.5) / 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(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (t_0 <= single(1.0))
		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((single(0.5) / 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) < -5

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. sub-neg 98.7%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 8.8%

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

      10. associate-*r/ 8.8%

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

      11. metadata-eval 8.8%

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

      12. distribute-neg-frac 8.8%

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

      13. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 1

   1. Initial program 26.8%

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

      1. +-commutative 26.8%

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

      2. hypot-1-def 26.8%

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

      3. flip-+ 26.5%

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

      4. hypot-1-def 26.5%

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

      5. hypot-1-def 26.4%

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

      6. add-sqr-sqrt 26.6%

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

      7. +-commutative 26.6%

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

      8. hypot-1-def 26.6%

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

      9. +-commutative 26.6%

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

      10. div-sub 26.6%

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

   4. Applied egg-rr 26.5%

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

      1. div-sub 26.5%

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

      2. remove-double-neg 26.5%

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

      3. distribute-frac-neg2 26.5%

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

      4. neg-sub0 26.5%

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

      5. associate--r- 26.5%

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

      6. neg-sub0 26.5%

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

      7. +-commutative 26.5%

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

      8. sub-neg 26.5%

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

      9. distribute-frac-neg 26.5%

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

      10. sub-neg 26.5%

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

      11. +-commutative 26.5%

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

      12. distribute-neg-in 26.5%

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

      13. remove-double-neg 26.5%

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

      14. fma-undefine 26.5%

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

      15. unpow2 26.5%

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

      16. +-commutative 26.5%

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

      17. associate-+l+ 26.6%

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

      18. sub-neg 26.6%

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

      19. +-inverses 26.6%

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

      20. metadata-eval 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in x around 0 99.2%

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

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

   1. Initial program 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 98.5%

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

      3. flip-+ 3.1%

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

      4. hypot-1-def 3.1%

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

      5. hypot-1-def 3.1%

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

      6. add-sqr-sqrt 2.9%

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

      7. +-commutative 2.9%

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

      8. hypot-1-def 2.9%

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

      9. +-commutative 2.9%

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

      10. div-sub 2.9%

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

   4. Applied egg-rr 2.9%

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

      1. div-sub 2.9%

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

      2. remove-double-neg 2.9%

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

      3. distribute-frac-neg2 2.9%

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

      4. neg-sub0 2.9%

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

      5. associate--r- 2.9%

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

      6. neg-sub0 2.9%

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

      7. +-commutative 2.9%

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

      8. sub-neg 2.9%

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

      9. distribute-frac-neg 2.9%

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

      10. sub-neg 2.9%

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

      11. +-commutative 2.9%

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

      12. distribute-neg-in 2.9%

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

      13. remove-double-neg 2.9%

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

      14. fma-undefine 2.9%

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

      15. unpow2 2.9%

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

      16. +-commutative 2.9%

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

      17. associate-+l+ 5.6%

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

      18. sub-neg 5.6%

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

      19. +-inverses 8.9%

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

      20. metadata-eval 8.9%

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

   6. Simplified 8.9%

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

      1. inv-pow 8.9%

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

      2. sub-neg 8.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 51.8%

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

      5. sqr-neg 51.8%

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

      6. sqrt-unprod 98.5%

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

      7. add-sqr-sqrt 98.5%

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

      8. +-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

      1. unpow-1 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.03999999910593033)
     (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 <= -100.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.03999999910593033f) {
		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(-100.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.03999999910593033))
		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(-100.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.03999999910593033))
		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 < -100

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. sub-neg 98.7%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 8.8%

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

      10. associate-*r/ 8.8%

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

      11. metadata-eval 8.8%

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

      12. distribute-neg-frac 8.8%

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

      13. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -100 < x < 0.0399999991

   1. Initial program 25.0%

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

      1. +-commutative 25.0%

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

      2. hypot-1-def 25.0%

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

      3. flip-+ 24.8%

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

      4. hypot-1-def 24.8%

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

      5. hypot-1-def 24.7%

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

      6. add-sqr-sqrt 25.0%

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

      7. +-commutative 25.0%

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

      8. hypot-1-def 25.0%

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

      9. +-commutative 25.0%

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

      10. div-sub 24.9%

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

   4. Applied egg-rr 24.8%

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

      1. div-sub 24.8%

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

      2. remove-double-neg 24.8%

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

      3. distribute-frac-neg2 24.8%

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

      4. neg-sub0 24.8%

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

      5. associate--r- 24.8%

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

      6. neg-sub0 24.8%

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

      7. +-commutative 24.8%

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

      8. sub-neg 24.8%

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

      9. distribute-frac-neg 24.8%

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

      10. sub-neg 24.8%

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

      11. +-commutative 24.8%

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

      12. distribute-neg-in 24.8%

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

      13. remove-double-neg 24.8%

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

      14. fma-undefine 24.8%

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

      15. unpow2 24.8%

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

      16. +-commutative 24.8%

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

      17. associate-+l+ 25.0%

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

      18. sub-neg 25.0%

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

      19. +-inverses 25.0%

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

      20. metadata-eval 25.0%

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

   6. Simplified 25.0%

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

   7. Taylor expanded in x around 0 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. unpow2 99.5%

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

      6. cube-mult 99.5%

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

   9. Simplified 99.5%

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

   if 0.0399999991 < x

   1. Initial program 53.8%

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

   3. Taylor expanded in x around 0 53.8%

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

      1. rem-square-sqrt 53.8%

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

      2. fabs-sqr 53.8%

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

      3. rem-square-sqrt 53.8%

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

      4. metadata-eval 53.8%

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

      5. unpow2 53.8%

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

      6. hypot-undefine 98.5%

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

   5. Simplified 98.5%

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

Alternative 4: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -100.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-100.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-100.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -100

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. sub-neg 98.7%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 8.8%

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

      10. associate-*r/ 8.8%

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

      11. metadata-eval 8.8%

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

      12. distribute-neg-frac 8.8%

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

      13. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -100 < x < 1

   1. Initial program 26.8%

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

      1. +-commutative 26.8%

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

      2. hypot-1-def 26.8%

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

      3. flip-+ 26.5%

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

      4. hypot-1-def 26.5%

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

      5. hypot-1-def 26.4%

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

      6. add-sqr-sqrt 26.6%

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

      7. +-commutative 26.6%

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

      8. hypot-1-def 26.6%

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

      9. +-commutative 26.6%

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

      10. div-sub 26.6%

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

   4. Applied egg-rr 26.5%

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

      1. div-sub 26.5%

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

      2. remove-double-neg 26.5%

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

      3. distribute-frac-neg2 26.5%

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

      4. neg-sub0 26.5%

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

      5. associate--r- 26.5%

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

      6. neg-sub0 26.5%

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

      7. +-commutative 26.5%

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

      8. sub-neg 26.5%

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

      9. distribute-frac-neg 26.5%

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

      10. sub-neg 26.5%

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

      11. +-commutative 26.5%

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

      12. distribute-neg-in 26.5%

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

      13. remove-double-neg 26.5%

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

      14. fma-undefine 26.5%

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

      15. unpow2 26.5%

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

      16. +-commutative 26.5%

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

      17. associate-+l+ 26.6%

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

      18. sub-neg 26.6%

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

      19. +-inverses 26.6%

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

      20. metadata-eval 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in x around 0 98.7%

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

      1. distribute-lft-in 98.7%

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

      2. *-rgt-identity 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-*r* 98.7%

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

      5. unpow2 98.7%

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

      6. cube-mult 98.7%

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

   9. Simplified 98.7%

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

   if 1 < x

   1. Initial program 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 98.5%

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

      3. flip-+ 3.1%

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

      4. hypot-1-def 3.1%

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

      5. hypot-1-def 3.1%

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

      6. add-sqr-sqrt 2.9%

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

      7. +-commutative 2.9%

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

      8. hypot-1-def 2.9%

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

      9. +-commutative 2.9%

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

      10. div-sub 2.9%

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

   4. Applied egg-rr 2.9%

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

      1. div-sub 2.9%

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

      2. remove-double-neg 2.9%

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

      3. distribute-frac-neg2 2.9%

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

      4. neg-sub0 2.9%

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

      5. associate--r- 2.9%

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

      6. neg-sub0 2.9%

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

      7. +-commutative 2.9%

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

      8. sub-neg 2.9%

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

      9. distribute-frac-neg 2.9%

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

      10. sub-neg 2.9%

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

      11. +-commutative 2.9%

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

      12. distribute-neg-in 2.9%

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

      13. remove-double-neg 2.9%

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

      14. fma-undefine 2.9%

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

      15. unpow2 2.9%

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

      16. +-commutative 2.9%

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

      17. associate-+l+ 5.6%

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

      18. sub-neg 5.6%

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

      19. +-inverses 8.9%

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

      20. metadata-eval 8.9%

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

   6. Simplified 8.9%

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

      1. inv-pow 8.9%

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

      2. sub-neg 8.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 51.8%

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

      5. sqr-neg 51.8%

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

      6. sqrt-unprod 98.5%

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

      7. add-sqr-sqrt 98.5%

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

      8. +-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

      1. unpow-1 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in x around inf 99.9%

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

Alternative 5: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -100.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-100.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-100.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -100

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. sub-neg 98.7%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 8.8%

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

      10. associate-*r/ 8.8%

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

      11. metadata-eval 8.8%

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

      12. distribute-neg-frac 8.8%

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

      13. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -100 < x < 1

   1. Initial program 26.8%

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

   3. Taylor expanded in x around 0 22.1%

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

      1. log1p-define 91.4%

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

      2. rem-square-sqrt 42.1%

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

      3. fabs-sqr 42.1%

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

      4. rem-square-sqrt 91.4%

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

   5. Simplified 91.4%

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

   6. Taylor expanded in x around 0 96.4%

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

   if 1 < x

   1. Initial program 51.8%

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

      1. +-commutative 51.8%

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

      2. hypot-1-def 98.5%

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

      3. flip-+ 3.1%

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

      4. hypot-1-def 3.1%

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

      5. hypot-1-def 3.1%

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

      6. add-sqr-sqrt 2.9%

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

      7. +-commutative 2.9%

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

      8. hypot-1-def 2.9%

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

      9. +-commutative 2.9%

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

      10. div-sub 2.9%

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

   4. Applied egg-rr 2.9%

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

      1. div-sub 2.9%

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

      2. remove-double-neg 2.9%

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

      3. distribute-frac-neg2 2.9%

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

      4. neg-sub0 2.9%

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

      5. associate--r- 2.9%

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

      6. neg-sub0 2.9%

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

      7. +-commutative 2.9%

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

      8. sub-neg 2.9%

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

      9. distribute-frac-neg 2.9%

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

      10. sub-neg 2.9%

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

      11. +-commutative 2.9%

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

      12. distribute-neg-in 2.9%

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

      13. remove-double-neg 2.9%

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

      14. fma-undefine 2.9%

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

      15. unpow2 2.9%

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

      16. +-commutative 2.9%

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

      17. associate-+l+ 5.6%

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

      18. sub-neg 5.6%

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

      19. +-inverses 8.9%

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

      20. metadata-eval 8.9%

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

   6. Simplified 8.9%

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

      1. inv-pow 8.9%

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

      2. sub-neg 8.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 51.8%

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

      5. sqr-neg 51.8%

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

      6. sqrt-unprod 98.5%

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

      7. add-sqr-sqrt 98.5%

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

      8. +-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

      1. unpow-1 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in x around inf 99.9%

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

Alternative 6: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -100.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-100.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-100.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -100

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. sub-neg 98.7%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 8.8%

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

      10. associate-*r/ 8.8%

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

      11. metadata-eval 8.8%

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

      12. distribute-neg-frac 8.8%

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

      13. metadata-eval 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around 0 99.5%

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

   if -100 < x < 1

   1. Initial program 26.8%

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

   3. Taylor expanded in x around 0 22.1%

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

      1. log1p-define 91.4%

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

      2. rem-square-sqrt 42.1%

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

      3. fabs-sqr 42.1%

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

      4. rem-square-sqrt 91.4%

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

   5. Simplified 91.4%

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

   6. Taylor expanded in x around 0 96.4%

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

   if 1 < x

   1. Initial program 51.8%

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

   3. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. rem-square-sqrt 98.5%

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

      3. fabs-sqr 98.5%

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

      4. rem-square-sqrt 98.5%

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

      5. *-inverses 98.5%

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

      6. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

Alternative 7: 96.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0)
   (copysign (log (* x -2.0)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -100.0f) {
		tmp = copysignf(logf((x * -2.0f)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-100.0))
		tmp = copysign(log(Float32(x * Float32(-2.0))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-100.0))
		tmp = sign(x) * abs(log((x * single(-2.0))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -100

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

      2. hypot-1-def 98.7%

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

      3. flip-+ 2.6%

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

      4. hypot-1-def 2.6%

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

      5. hypot-1-def 2.5%

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

      6. add-sqr-sqrt 2.6%

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

      7. +-commutative 2.6%

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

      8. hypot-1-def 2.6%

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

      9. +-commutative 2.6%

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

      10. div-sub 2.7%

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

   4. Applied egg-rr 5.8%

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

      1. div-sub 7.2%

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

      2. remove-double-neg 7.2%

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

      3. distribute-frac-neg2 7.2%

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

      4. neg-sub0 7.2%

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

      5. associate--r- 7.2%

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

      6. neg-sub0 7.2%

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

      7. +-commutative 7.2%

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

      8. sub-neg 7.2%

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

      9. distribute-frac-neg 7.2%

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

      10. sub-neg 7.2%

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

      11. +-commutative 7.2%

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

      12. distribute-neg-in 7.2%

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

      13. remove-double-neg 7.2%

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

      14. fma-undefine 7.2%

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

      15. unpow2 7.2%

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

      16. +-commutative 7.2%

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

      17. associate-+l+ 50.0%

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

      18. sub-neg 50.0%

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

      19. +-inverses 98.6%

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

      20. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

      1. inv-pow 98.6%

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

      2. sub-neg 98.6%

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

      3. add-sqr-sqrt 98.7%

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

      4. sqrt-unprod 53.2%

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

      5. sqr-neg 53.2%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 8.7%

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

      8. +-commutative 8.7%

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

   8. Applied egg-rr 8.7%

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

      1. unpow-1 8.7%

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

   10. Simplified 8.7%

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

   11. Taylor expanded in x around -inf 98.2%

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

       1. *-commutative 98.2%

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

   13. Simplified 98.2%

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

   if -100 < x < 1

   1. Initial program 26.8%

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

   3. Taylor expanded in x around 0 22.1%

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

      1. log1p-define 91.4%

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

      2. rem-square-sqrt 42.1%

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

      3. fabs-sqr 42.1%

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

      4. rem-square-sqrt 91.4%

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

   5. Simplified 91.4%

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

   6. Taylor expanded in x around 0 96.4%

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

   if 1 < x

   1. Initial program 51.8%

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

   3. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. rem-square-sqrt 98.5%

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

      3. fabs-sqr 98.5%

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

      4. rem-square-sqrt 98.5%

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

      5. *-inverses 98.5%

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

      6. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

Alternative 8: 81.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0) (copysign (log (* x -2.0)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -100.0f) {
		tmp = copysignf(logf((x * -2.0f)), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-100.0))
		tmp = copysign(log(Float32(x * Float32(-2.0))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -100

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

      2. hypot-1-def 98.7%

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

      3. flip-+ 2.6%

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

      4. hypot-1-def 2.6%

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

      5. hypot-1-def 2.5%

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

      6. add-sqr-sqrt 2.6%

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

      7. +-commutative 2.6%

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

      8. hypot-1-def 2.6%

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

      9. +-commutative 2.6%

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

      10. div-sub 2.7%

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

   4. Applied egg-rr 5.8%

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

      1. div-sub 7.2%

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

      2. remove-double-neg 7.2%

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

      3. distribute-frac-neg2 7.2%

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

      4. neg-sub0 7.2%

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

      5. associate--r- 7.2%

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

      6. neg-sub0 7.2%

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

      7. +-commutative 7.2%

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

      8. sub-neg 7.2%

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

      9. distribute-frac-neg 7.2%

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

      10. sub-neg 7.2%

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

      11. +-commutative 7.2%

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

      12. distribute-neg-in 7.2%

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

      13. remove-double-neg 7.2%

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

      14. fma-undefine 7.2%

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

      15. unpow2 7.2%

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

      16. +-commutative 7.2%

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

      17. associate-+l+ 50.0%

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

      18. sub-neg 50.0%

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

      19. +-inverses 98.6%

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

      20. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

      1. inv-pow 98.6%

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

      2. sub-neg 98.6%

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

      3. add-sqr-sqrt 98.7%

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

      4. sqrt-unprod 53.2%

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

      5. sqr-neg 53.2%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 8.7%

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

      8. +-commutative 8.7%

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

   8. Applied egg-rr 8.7%

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

      1. unpow-1 8.7%

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

   10. Simplified 8.7%

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

   11. Taylor expanded in x around -inf 98.2%

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

       1. *-commutative 98.2%

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

   13. Simplified 98.2%

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

   if -100 < x

   1. Initial program 35.4%

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

   3. Taylor expanded in x around 0 29.8%

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

      1. log1p-define 75.3%

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

      2. rem-square-sqrt 42.9%

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

      3. fabs-sqr 42.9%

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

      4. rem-square-sqrt 75.3%

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

   5. Simplified 75.3%

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

Alternative 9: 68.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -100.0) (copysign (log (- x)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -100.0f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-100.0))
		tmp = copysign(log(Float32(-x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -100

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 44.9%

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

      1. neg-mul-1 44.9%

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

   5. Simplified 44.9%

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

   if -100 < x

   1. Initial program 35.4%

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

   3. Taylor expanded in x around 0 29.8%

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

      1. log1p-define 75.3%

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

      2. rem-square-sqrt 42.9%

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

      3. fabs-sqr 42.9%

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

      4. rem-square-sqrt 75.3%

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

   5. Simplified 75.3%

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

Alternative 10: 61.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 1.0) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 36.5%

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

   3. Taylor expanded in x around 0 30.4%

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

      1. log1p-define 74.5%

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

      2. rem-square-sqrt 26.7%

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

      3. fabs-sqr 26.7%

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

      4. rem-square-sqrt 58.1%

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

   5. Simplified 58.1%

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

   6. Taylor expanded in x around 0 65.1%

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

   if 1 < x

   1. Initial program 51.8%

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

   3. Taylor expanded in x around 0 44.5%

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

      1. log1p-define 44.5%

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

      2. rem-square-sqrt 44.5%

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

      3. fabs-sqr 44.5%

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

      4. rem-square-sqrt 44.5%

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

   5. Simplified 44.5%

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

Alternative 11: 53.6% 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 40.3%

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

3. Taylor expanded in x around 0 33.9%

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

   1. log1p-define 67.0%

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

   2. rem-square-sqrt 31.2%

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

   3. fabs-sqr 31.2%

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

   4. rem-square-sqrt 54.7%

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

5. Simplified 54.7%

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

6. Taylor expanded in x around 0 51.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
7. 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 2024137 
(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))