Rust f32::asinh

Percentage Accurate: 37.0% → 99.0%

Time: 7.2s

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: 37.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (log (- (hypot 1.0 x) 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 <= -0.20000000298023224f) {
		tmp = copysignf(logf((hypotf(1.0f, x) - 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(-0.20000000298023224))
		tmp = copysign(log(Float32(hypot(Float32(1.0), x) - 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(Float32(-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(-0.20000000298023224))
		tmp = sign(x) * abs(log((hypot(single(1.0), x) - 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) < -0.200000003

   1. Initial program 53.9%

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

      1. flip-+ 7.0%

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

      2. clear-num 7.1%

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

      3. log-div 7.1%

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

      4. metadata-eval 7.1%

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

      5. +-commutative 7.1%

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

      6. hypot-1-def 7.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 8.9%

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

   4. Applied egg-rr 9.5%

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

      1. neg-sub0 9.5%

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

      2. div-sub 9.5%

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

      3. *-rgt-identity 9.5%

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

      4. associate-/l* 9.5%

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

      5. *-rgt-identity 9.5%

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

      6. associate-/l* 9.5%

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

      7. fma-undefine 8.9%

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

      8. unpow2 8.9%

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

      9. associate--r+ 9.6%

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

      10. +-inverses 9.6%

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

      11. metadata-eval 9.6%

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

      12. metadata-eval 9.6%

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

      13. *-commutative 9.6%

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

      14. neg-mul-1 9.6%

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

      15. neg-sub0 9.6%

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

      16. fma-undefine 8.9%

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

      17. unpow2 8.9%

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

   6. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 99.8%

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

      4. sqr-neg 99.8%

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

      5. sqrt-unprod 98.5%

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

      6. add-sqr-sqrt 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

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

   1. Initial program 24.0%

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

      1. +-commutative 24.0%

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

      2. hypot-1-def 24.1%

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

      3. flip-+ 24.0%

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

      4. hypot-1-def 24.1%

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

      5. hypot-1-def 24.0%

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

      6. add-sqr-sqrt 24.0%

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

      7. +-commutative 24.0%

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

      8. hypot-1-def 23.9%

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

      9. +-commutative 23.9%

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

      10. div-sub 24.0%

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

   4. Applied egg-rr 23.9%

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

      1. div-sub 24.0%

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

      2. remove-double-neg 24.0%

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

      3. distribute-frac-neg2 24.0%

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

      4. distribute-frac-neg 24.0%

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

      5. neg-sub0 24.0%

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

      6. associate--r- 24.0%

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

      7. neg-sub0 24.0%

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

      8. +-commutative 24.0%

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

      9. fma-undefine 24.0%

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

      10. unpow2 24.0%

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

      11. +-commutative 24.0%

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

      12. associate-+l+ 23.9%

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

      13. sub-neg 23.9%

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

      14. +-inverses 23.9%

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

      15. metadata-eval 23.9%

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

      16. neg-sub0 23.9%

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

      17. associate--r- 23.9%

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

      18. neg-sub0 23.9%

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

      19. +-commutative 23.9%

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

      20. sub-neg 23.9%

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

   6. Simplified 23.9%

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

   7. Taylor expanded in x around 0 99.7%

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

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

   1. Initial program 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. flip-+ 3.6%

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

      2. clear-num 3.6%

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

      3. log-div 3.6%

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

      4. metadata-eval 3.6%

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

      5. +-commutative 3.6%

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

      6. hypot-1-def 3.4%

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

      7. add-sqr-sqrt 4.1%

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

      8. fabs-sqr 4.1%

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

      9. add-sqr-sqrt 3.4%

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

   4. Applied egg-rr 3.5%

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

      1. neg-sub0 3.5%

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

      2. div-sub 3.5%

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

      3. *-rgt-identity 3.5%

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

      4. associate-/l* 3.5%

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

      5. *-rgt-identity 3.5%

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

      6. associate-/l* 3.5%

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

      7. fma-undefine 3.5%

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

      8. unpow2 3.5%

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

      9. associate--r+ 3.5%

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

      10. +-inverses 3.5%

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

      11. metadata-eval 3.5%

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

      12. metadata-eval 3.5%

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

      13. *-commutative 3.5%

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

      14. neg-mul-1 3.5%

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

      15. neg-sub0 3.5%

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

      16. fma-undefine 3.5%

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

      17. unpow2 3.5%

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

   6. Simplified 9.7%

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

   7. Taylor expanded in x around inf 98.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.11999999731779099:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + \mathsf{fma}\left(x \cdot x, 0.075, -0.16666666666666666\right) \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.11999999731779099)
     (copysign (log (- (hypot 1.0 x) x)) x)
     (if (<= t_0 1.0)
       (copysign
        (+ x (* (fma (* x x) 0.075 -0.16666666666666666) (pow x 3.0)))
        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 <= -0.11999999731779099f) {
		tmp = copysignf(logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 1.0f) {
		tmp = copysignf((x + (fmaf((x * x), 0.075f, -0.16666666666666666f) * powf(x, 3.0f))), 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(-0.11999999731779099))
		tmp = copysign(log(Float32(hypot(Float32(1.0), x) - x)), x);
	elseif (t_0 <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(fma(Float32(x * x), Float32(0.075), Float32(-0.16666666666666666)) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -0.119999997

   1. Initial program 54.7%

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

      1. flip-+ 8.6%

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

      2. clear-num 8.7%

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

      3. log-div 8.7%

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

      4. metadata-eval 8.7%

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

      5. +-commutative 8.7%

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

      6. hypot-1-def 8.6%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 10.4%

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

   4. Applied egg-rr 11.1%

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

      1. neg-sub0 11.1%

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

      2. div-sub 11.1%

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

      3. *-rgt-identity 11.1%

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

      4. associate-/l* 11.1%

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

      5. *-rgt-identity 11.1%

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

      6. associate-/l* 11.1%

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

      7. fma-undefine 10.5%

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

      8. unpow2 10.5%

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

      9. associate--r+ 11.2%

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

      10. +-inverses 11.2%

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

      11. metadata-eval 11.2%

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

      12. metadata-eval 11.2%

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

      13. *-commutative 11.2%

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

      14. neg-mul-1 11.2%

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

      15. neg-sub0 11.2%

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

      16. fma-undefine 10.5%

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

      17. unpow2 10.5%

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

   6. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 99.7%

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

      4. sqr-neg 99.7%

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

      5. sqrt-unprod 98.5%

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

      6. add-sqr-sqrt 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. *-lft-identity 99.7%

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

   10. Simplified 99.7%

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

   if -0.119999997 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 1

   1. Initial program 23.5%

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

      1. +-commutative 23.5%

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

      2. hypot-1-def 23.5%

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

      3. flip-+ 23.5%

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

      4. hypot-1-def 23.6%

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

      5. hypot-1-def 23.4%

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

      6. add-sqr-sqrt 23.5%

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

      7. +-commutative 23.5%

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

      8. hypot-1-def 23.4%

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

      9. +-commutative 23.4%

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

      10. div-sub 23.4%

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

   4. Applied egg-rr 23.4%

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

      1. div-sub 23.4%

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

      2. remove-double-neg 23.4%

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

      3. distribute-frac-neg2 23.4%

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

      4. distribute-frac-neg 23.4%

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

      5. neg-sub0 23.4%

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

      6. associate--r- 23.4%

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

      7. neg-sub0 23.4%

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

      8. +-commutative 23.4%

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

      9. fma-undefine 23.4%

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

      10. unpow2 23.4%

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

      11. +-commutative 23.4%

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

      12. associate-+l+ 23.4%

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

      13. sub-neg 23.4%

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

      14. +-inverses 23.4%

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

      15. metadata-eval 23.4%

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

      16. neg-sub0 23.4%

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

      17. associate--r- 23.4%

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

      18. neg-sub0 23.4%

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

      19. +-commutative 23.4%

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

      20. sub-neg 23.4%

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

   6. Simplified 23.4%

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

   7. Taylor expanded in x around 0 99.6%

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

      1. distribute-rgt-in 99.7%

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

      2. *-lft-identity 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. unpow2 99.7%

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

      9. unpow3 99.7%

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

   9. Simplified 99.7%

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

       1. unpow2 99.7%

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

   11. Applied egg-rr 99.7%

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

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

   1. Initial program 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. flip-+ 3.6%

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

      2. clear-num 3.6%

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

      3. log-div 3.6%

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

      4. metadata-eval 3.6%

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

      5. +-commutative 3.6%

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

      6. hypot-1-def 3.4%

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

      7. add-sqr-sqrt 4.1%

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

      8. fabs-sqr 4.1%

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

      9. add-sqr-sqrt 3.4%

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

   4. Applied egg-rr 3.5%

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

      1. neg-sub0 3.5%

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

      2. div-sub 3.5%

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

      3. *-rgt-identity 3.5%

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

      4. associate-/l* 3.5%

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

      5. *-rgt-identity 3.5%

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

      6. associate-/l* 3.5%

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

      7. fma-undefine 3.5%

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

      8. unpow2 3.5%

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

      9. associate--r+ 3.5%

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

      10. +-inverses 3.5%

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

      11. metadata-eval 3.5%

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

      12. metadata-eval 3.5%

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

      13. *-commutative 3.5%

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

      14. neg-mul-1 3.5%

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

      15. neg-sub0 3.5%

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

      16. fma-undefine 3.5%

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

      17. unpow2 3.5%

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

   6. Simplified 9.7%

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

   7. Taylor expanded in x around inf 98.9%

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

Alternative 3: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.10000000149011612)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (- (log (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.10000000149011612f) {
		tmp = copysignf(logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(-logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.10000000149011612))
		tmp = copysign(log(Float32(hypot(Float32(1.0), x) - x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.10000000149011612))
		tmp = sign(x) * abs(log((hypot(single(1.0), x) - x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	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 < -0.100000001

   1. Initial program 55.4%

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

      1. flip-+ 10.2%

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

      2. clear-num 10.2%

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

      3. log-div 10.2%

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

      4. metadata-eval 10.2%

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

      5. +-commutative 10.2%

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

      6. hypot-1-def 10.1%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 11.8%

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

   4. Applied egg-rr 12.6%

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

      1. neg-sub0 12.6%

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

      2. div-sub 12.6%

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

      3. *-rgt-identity 12.6%

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

      4. associate-/l* 12.6%

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

      5. *-rgt-identity 12.6%

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

      6. associate-/l* 12.6%

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

      7. fma-undefine 12.0%

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

      8. unpow2 12.0%

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

      9. associate--r+ 12.7%

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

      10. +-inverses 12.7%

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

      11. metadata-eval 12.7%

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

      12. metadata-eval 12.7%

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

      13. *-commutative 12.7%

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

      14. neg-mul-1 12.7%

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

      15. neg-sub0 12.7%

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

      16. fma-undefine 12.0%

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

      17. unpow2 12.0%

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

   6. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 99.7%

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

      4. sqr-neg 99.7%

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

      5. sqrt-unprod 98.5%

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

      6. add-sqr-sqrt 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. *-lft-identity 99.7%

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

   10. Simplified 99.7%

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

   if -0.100000001 < x < 1

   1. Initial program 23.0%

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

      1. +-commutative 23.0%

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

      2. hypot-1-def 23.0%

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

      3. flip-+ 22.9%

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

      4. hypot-1-def 23.0%

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

      5. hypot-1-def 22.9%

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

      6. add-sqr-sqrt 23.0%

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

      7. +-commutative 23.0%

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

      8. hypot-1-def 22.9%

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

      9. +-commutative 22.9%

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

      10. div-sub 23.0%

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

   4. Applied egg-rr 22.9%

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

      1. div-sub 22.9%

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

      2. remove-double-neg 22.9%

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

      3. distribute-frac-neg2 22.9%

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

      4. distribute-frac-neg 22.9%

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

      5. neg-sub0 22.9%

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

      6. associate--r- 22.9%

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

      7. neg-sub0 22.9%

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

      8. +-commutative 22.9%

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

      9. fma-undefine 22.9%

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

      10. unpow2 22.9%

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

      11. +-commutative 22.9%

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

      12. associate-+l+ 22.9%

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

      13. sub-neg 22.9%

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

      14. +-inverses 22.9%

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

      15. metadata-eval 22.9%

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

      16. neg-sub0 22.9%

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

      17. associate--r- 22.9%

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

      18. neg-sub0 22.9%

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

      19. +-commutative 22.9%

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

      20. sub-neg 22.9%

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

   6. Simplified 22.9%

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

   7. Taylor expanded in x around 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-rgt-in 99.5%

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

      2. *-lft-identity 99.5%

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

      3. associate-*l* 99.5%

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

      4. unpow2 99.5%

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

      5. unpow3 99.5%

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

   9. Simplified 99.5%

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

   if 1 < x

   1. Initial program 53.2%

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

      1. flip-+ 3.6%

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

      2. clear-num 3.6%

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

      3. log-div 3.6%

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

      4. metadata-eval 3.6%

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

      5. +-commutative 3.6%

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

      6. hypot-1-def 3.4%

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

      7. add-sqr-sqrt 4.1%

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

      8. fabs-sqr 4.1%

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

      9. add-sqr-sqrt 3.4%

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

   4. Applied egg-rr 3.5%

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

      1. neg-sub0 3.5%

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

      2. div-sub 3.5%

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

      3. *-rgt-identity 3.5%

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

      4. associate-/l* 3.5%

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

      5. *-rgt-identity 3.5%

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

      6. associate-/l* 3.5%

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

      7. fma-undefine 3.5%

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

      8. unpow2 3.5%

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

      9. associate--r+ 3.5%

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

      10. +-inverses 3.5%

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

      11. metadata-eval 3.5%

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

      12. metadata-eval 3.5%

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

      13. *-commutative 3.5%

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

      14. neg-mul-1 3.5%

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

      15. neg-sub0 3.5%

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

      16. fma-undefine 3.5%

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

      17. unpow2 3.5%

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

   6. Simplified 9.7%

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

   7. Taylor expanded in x around inf 98.9%

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

Alternative 4: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -500.0)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (- (log (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -500.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(-logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-500.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-500.0))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	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 < -500

   1. Initial program 51.4%

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

      1. +-commutative 51.4%

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

      2. hypot-1-def 100.0%

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

      3. flip-+ 1.9%

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

      4. hypot-1-def 1.9%

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

      5. hypot-1-def 1.9%

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

      6. add-sqr-sqrt 1.9%

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

      7. +-commutative 1.9%

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

      8. hypot-1-def 1.9%

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

      9. +-commutative 1.9%

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

      10. div-sub 1.8%

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

   4. Applied egg-rr 5.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
   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. distribute-frac-neg 7.2%

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

      5. neg-sub0 7.2%

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

      6. associate--r- 7.2%

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

      7. neg-sub0 7.2%

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

      8. +-commutative 7.2%

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

      9. fma-undefine 6.6%

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

      10. unpow2 6.6%

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

      11. +-commutative 6.6%

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

      12. associate-+l+ 48.0%

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

      13. sub-neg 48.0%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around -inf 99.9%

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

      1. clear-num 99.9%

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

      2. log-rec 99.9%

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

      3. div-inv 99.9%

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

      4. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -500 < x < 1

   1. Initial program 25.5%

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

      1. +-commutative 25.5%

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

      2. hypot-1-def 25.6%

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

      3. flip-+ 25.5%

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

      4. hypot-1-def 25.6%

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

      5. hypot-1-def 25.4%

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

      6. add-sqr-sqrt 25.6%

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

      7. +-commutative 25.6%

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

      8. hypot-1-def 25.4%

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

      9. +-commutative 25.4%

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

      10. div-sub 25.4%

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

   4. Applied egg-rr 25.4%

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

      1. div-sub 25.5%

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

      2. remove-double-neg 25.5%

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

      3. distribute-frac-neg2 25.5%

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

      4. distribute-frac-neg 25.5%

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

      5. neg-sub0 25.5%

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

      6. associate--r- 25.5%

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

      7. neg-sub0 25.5%

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

      8. +-commutative 25.5%

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

      9. fma-undefine 25.5%

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

      10. unpow2 25.5%

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

      11. +-commutative 25.5%

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

      12. associate-+l+ 25.4%

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

      13. sub-neg 25.4%

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

      14. +-inverses 25.4%

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

      15. metadata-eval 25.4%

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

      16. neg-sub0 25.4%

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

      17. associate--r- 25.4%

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

      18. neg-sub0 25.4%

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

      19. +-commutative 25.4%

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

      20. sub-neg 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in x around 0 98.2%

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

      1. distribute-rgt-in 98.2%

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

      2. *-lft-identity 98.2%

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

      3. associate-*l* 98.2%

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

      4. unpow2 98.2%

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

      5. unpow3 98.2%

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

   9. Simplified 98.2%

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

   if 1 < x

   1. Initial program 53.2%

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

      1. flip-+ 3.6%

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

      2. clear-num 3.6%

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

      3. log-div 3.6%

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

      4. metadata-eval 3.6%

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

      5. +-commutative 3.6%

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

      6. hypot-1-def 3.4%

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

      7. add-sqr-sqrt 4.1%

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

      8. fabs-sqr 4.1%

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

      9. add-sqr-sqrt 3.4%

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

   4. Applied egg-rr 3.5%

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

      1. neg-sub0 3.5%

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

      2. div-sub 3.5%

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

      3. *-rgt-identity 3.5%

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

      4. associate-/l* 3.5%

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

      5. *-rgt-identity 3.5%

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

      6. associate-/l* 3.5%

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

      7. fma-undefine 3.5%

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

      8. unpow2 3.5%

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

      9. associate--r+ 3.5%

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

      10. +-inverses 3.5%

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

      11. metadata-eval 3.5%

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

      12. metadata-eval 3.5%

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

      13. *-commutative 3.5%

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

      14. neg-mul-1 3.5%

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

      15. neg-sub0 3.5%

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

      16. fma-undefine 3.5%

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

      17. unpow2 3.5%

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

   6. Simplified 9.7%

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

   7. Taylor expanded in x around inf 98.9%

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

Alternative 5: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -500.0)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 1.0)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (- (log (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -500.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(-logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-500.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-500.0))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * 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 < -500

   1. Initial program 51.4%

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

      1. +-commutative 51.4%

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

      2. hypot-1-def 100.0%

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

      3. flip-+ 1.9%

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

      4. hypot-1-def 1.9%

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

      5. hypot-1-def 1.9%

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

      6. add-sqr-sqrt 1.9%

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

      7. +-commutative 1.9%

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

      8. hypot-1-def 1.9%

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

      9. +-commutative 1.9%

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

      10. div-sub 1.8%

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

   4. Applied egg-rr 5.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
   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. distribute-frac-neg 7.2%

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

      5. neg-sub0 7.2%

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

      6. associate--r- 7.2%

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

      7. neg-sub0 7.2%

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

      8. +-commutative 7.2%

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

      9. fma-undefine 6.6%

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

      10. unpow2 6.6%

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

      11. +-commutative 6.6%

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

      12. associate-+l+ 48.0%

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

      13. sub-neg 48.0%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around -inf 99.9%

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

      1. clear-num 99.9%

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

      2. log-rec 99.9%

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

      3. div-inv 99.9%

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

      4. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -500 < x < 1

   1. Initial program 25.5%

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

      1. +-commutative 25.5%

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

      2. hypot-1-def 25.6%

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

      3. flip-+ 25.5%

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

      4. hypot-1-def 25.6%

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

      5. hypot-1-def 25.4%

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

      6. add-sqr-sqrt 25.6%

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

      7. +-commutative 25.6%

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

      8. hypot-1-def 25.4%

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

      9. +-commutative 25.4%

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

      10. div-sub 25.4%

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

   4. Applied egg-rr 25.4%

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

      1. div-sub 25.5%

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

      2. remove-double-neg 25.5%

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

      3. distribute-frac-neg2 25.5%

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

      4. distribute-frac-neg 25.5%

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

      5. neg-sub0 25.5%

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

      6. associate--r- 25.5%

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

      7. neg-sub0 25.5%

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

      8. +-commutative 25.5%

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

      9. fma-undefine 25.5%

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

      10. unpow2 25.5%

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

      11. +-commutative 25.5%

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

      12. associate-+l+ 25.4%

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

      13. sub-neg 25.4%

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

      14. +-inverses 25.4%

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

      15. metadata-eval 25.4%

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

      16. neg-sub0 25.4%

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

      17. associate--r- 25.4%

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

      18. neg-sub0 25.4%

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

      19. +-commutative 25.4%

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

      20. sub-neg 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   9. Simplified 98.2%

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

       1. unpow2 98.9%

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

   11. Applied egg-rr 98.2%

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

   if 1 < x

   1. Initial program 53.2%

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

      1. flip-+ 3.6%

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

      2. clear-num 3.6%

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

      3. log-div 3.6%

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

      4. metadata-eval 3.6%

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

      5. +-commutative 3.6%

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

      6. hypot-1-def 3.4%

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

      7. add-sqr-sqrt 4.1%

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

      8. fabs-sqr 4.1%

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

      9. add-sqr-sqrt 3.4%

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

   4. Applied egg-rr 3.5%

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

      1. neg-sub0 3.5%

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

      2. div-sub 3.5%

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

      3. *-rgt-identity 3.5%

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

      4. associate-/l* 3.5%

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

      5. *-rgt-identity 3.5%

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

      6. associate-/l* 3.5%

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

      7. fma-undefine 3.5%

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

      8. unpow2 3.5%

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

      9. associate--r+ 3.5%

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

      10. +-inverses 3.5%

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

      11. metadata-eval 3.5%

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

      12. metadata-eval 3.5%

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

      13. *-commutative 3.5%

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

      14. neg-mul-1 3.5%

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

      15. neg-sub0 3.5%

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

      16. fma-undefine 3.5%

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

      17. unpow2 3.5%

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

   6. Simplified 9.7%

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

   7. Taylor expanded in x around inf 98.9%

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

Alternative 6: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -500.0)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 1.0)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -500.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-500.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-500.0))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * single(-0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -500

   1. Initial program 51.4%

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

      1. +-commutative 51.4%

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

      2. hypot-1-def 100.0%

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

      3. flip-+ 1.9%

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

      4. hypot-1-def 1.9%

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

      5. hypot-1-def 1.9%

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

      6. add-sqr-sqrt 1.9%

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

      7. +-commutative 1.9%

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

      8. hypot-1-def 1.9%

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

      9. +-commutative 1.9%

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

      10. div-sub 1.8%

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

   4. Applied egg-rr 5.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
   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. distribute-frac-neg 7.2%

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

      5. neg-sub0 7.2%

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

      6. associate--r- 7.2%

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

      7. neg-sub0 7.2%

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

      8. +-commutative 7.2%

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

      9. fma-undefine 6.6%

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

      10. unpow2 6.6%

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

      11. +-commutative 6.6%

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

      12. associate-+l+ 48.0%

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

      13. sub-neg 48.0%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around -inf 99.9%

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

      1. clear-num 99.9%

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

      2. log-rec 99.9%

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

      3. div-inv 99.9%

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

      4. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -500 < x < 1

   1. Initial program 25.5%

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

      1. +-commutative 25.5%

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

      2. hypot-1-def 25.6%

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

      3. flip-+ 25.5%

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

      4. hypot-1-def 25.6%

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

      5. hypot-1-def 25.4%

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

      6. add-sqr-sqrt 25.6%

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

      7. +-commutative 25.6%

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

      8. hypot-1-def 25.4%

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

      9. +-commutative 25.4%

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

      10. div-sub 25.4%

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

   4. Applied egg-rr 25.4%

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

      1. div-sub 25.5%

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

      2. remove-double-neg 25.5%

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

      3. distribute-frac-neg2 25.5%

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

      4. distribute-frac-neg 25.5%

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

      5. neg-sub0 25.5%

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

      6. associate--r- 25.5%

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

      7. neg-sub0 25.5%

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

      8. +-commutative 25.5%

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

      9. fma-undefine 25.5%

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

      10. unpow2 25.5%

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

      11. +-commutative 25.5%

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

      12. associate-+l+ 25.4%

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

      13. sub-neg 25.4%

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

      14. +-inverses 25.4%

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

      15. metadata-eval 25.4%

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

      16. neg-sub0 25.4%

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

      17. associate--r- 25.4%

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

      18. neg-sub0 25.4%

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

      19. +-commutative 25.4%

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

      20. sub-neg 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   9. Simplified 98.2%

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

       1. unpow2 98.9%

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

   11. Applied egg-rr 98.2%

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

   if 1 < x

   1. Initial program 53.2%

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

   3. Taylor expanded in x around inf 95.8%

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

      1. rem-square-sqrt 95.7%

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

      2. fabs-sqr 95.7%

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

      3. rem-square-sqrt 95.8%

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

      4. *-inverses 95.8%

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

      5. metadata-eval 95.8%

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

   5. Simplified 95.8%

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

Alternative 7: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -500.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -500.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-500.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-500.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * single(-0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -500

   1. Initial program 51.4%

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

      1. +-commutative 51.4%

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

      2. hypot-1-def 100.0%

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

      3. flip-+ 1.9%

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

      4. hypot-1-def 1.9%

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

      5. hypot-1-def 1.9%

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

      6. add-sqr-sqrt 1.9%

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

      7. +-commutative 1.9%

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

      8. hypot-1-def 1.9%

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

      9. +-commutative 1.9%

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

      10. div-sub 1.8%

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

   4. Applied egg-rr 5.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
   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. distribute-frac-neg 7.2%

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

      5. neg-sub0 7.2%

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

      6. associate--r- 7.2%

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

      7. neg-sub0 7.2%

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

      8. +-commutative 7.2%

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

      9. fma-undefine 6.6%

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

      10. unpow2 6.6%

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

      11. +-commutative 6.6%

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

      12. associate-+l+ 48.0%

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

      13. sub-neg 48.0%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. neg-sub0 99.9%

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

      17. associate--r- 99.9%

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

      18. neg-sub0 99.9%

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

      19. +-commutative 99.9%

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

      20. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around -inf 99.9%

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

   if -500 < x < 1

   1. Initial program 25.5%

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

      1. +-commutative 25.5%

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

      2. hypot-1-def 25.6%

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

      3. flip-+ 25.5%

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

      4. hypot-1-def 25.6%

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

      5. hypot-1-def 25.4%

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

      6. add-sqr-sqrt 25.6%

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

      7. +-commutative 25.6%

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

      8. hypot-1-def 25.4%

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

      9. +-commutative 25.4%

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

      10. div-sub 25.4%

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

   4. Applied egg-rr 25.4%

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

      1. div-sub 25.5%

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

      2. remove-double-neg 25.5%

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

      3. distribute-frac-neg2 25.5%

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

      4. distribute-frac-neg 25.5%

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

      5. neg-sub0 25.5%

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

      6. associate--r- 25.5%

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

      7. neg-sub0 25.5%

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

      8. +-commutative 25.5%

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

      9. fma-undefine 25.5%

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

      10. unpow2 25.5%

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

      11. +-commutative 25.5%

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

      12. associate-+l+ 25.4%

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

      13. sub-neg 25.4%

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

      14. +-inverses 25.4%

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

      15. metadata-eval 25.4%

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

      16. neg-sub0 25.4%

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

      17. associate--r- 25.4%

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

      18. neg-sub0 25.4%

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

      19. +-commutative 25.4%

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

      20. sub-neg 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   9. Simplified 98.2%

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

       1. unpow2 98.9%

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

   11. Applied egg-rr 98.2%

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

   if 1 < x

   1. Initial program 53.2%

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

   3. Taylor expanded in x around inf 95.8%

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

      1. rem-square-sqrt 95.7%

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

      2. fabs-sqr 95.7%

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

      3. rem-square-sqrt 95.8%

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

      4. *-inverses 95.8%

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

      5. metadata-eval 95.8%

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

   5. Simplified 95.8%

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

Alternative 8: 84.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -500.0)
   (copysign (log (- x)) x)
   (if (<= x 1.0)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -500.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-500.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-500.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * single(-0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -500

   1. Initial program 51.4%

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

   3. Taylor expanded in x around -inf 44.5%

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

      1. neg-mul-1 44.5%

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

   5. Simplified 44.5%

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

   if -500 < x < 1

   1. Initial program 25.5%

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

      1. +-commutative 25.5%

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

      2. hypot-1-def 25.6%

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

      3. flip-+ 25.5%

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

      4. hypot-1-def 25.6%

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

      5. hypot-1-def 25.4%

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

      6. add-sqr-sqrt 25.6%

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

      7. +-commutative 25.6%

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

      8. hypot-1-def 25.4%

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

      9. +-commutative 25.4%

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

      10. div-sub 25.4%

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

   4. Applied egg-rr 25.4%

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

      1. div-sub 25.5%

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

      2. remove-double-neg 25.5%

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

      3. distribute-frac-neg2 25.5%

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

      4. distribute-frac-neg 25.5%

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

      5. neg-sub0 25.5%

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

      6. associate--r- 25.5%

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

      7. neg-sub0 25.5%

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

      8. +-commutative 25.5%

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

      9. fma-undefine 25.5%

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

      10. unpow2 25.5%

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

      11. +-commutative 25.5%

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

      12. associate-+l+ 25.4%

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

      13. sub-neg 25.4%

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

      14. +-inverses 25.4%

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

      15. metadata-eval 25.4%

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

      16. neg-sub0 25.4%

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

      17. associate--r- 25.4%

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

      18. neg-sub0 25.4%

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

      19. +-commutative 25.4%

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

      20. sub-neg 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   9. Simplified 98.2%

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

       1. unpow2 98.9%

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

   11. Applied egg-rr 98.2%

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

   if 1 < x

   1. Initial program 53.2%

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

   3. Taylor expanded in x around inf 95.8%

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

      1. rem-square-sqrt 95.7%

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

      2. fabs-sqr 95.7%

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

      3. rem-square-sqrt 95.8%

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

      4. *-inverses 95.8%

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

      5. metadata-eval 95.8%

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

   5. Simplified 95.8%

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

Alternative 9: 71.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -500.0)
   (copysign (log (- x)) x)
   (if (<= x 1.0)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log1p x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -500.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-500.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -500

   1. Initial program 51.4%

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

   3. Taylor expanded in x around -inf 44.5%

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

      1. neg-mul-1 44.5%

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

   5. Simplified 44.5%

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

   if -500 < x < 1

   1. Initial program 25.5%

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

      1. +-commutative 25.5%

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

      2. hypot-1-def 25.6%

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

      3. flip-+ 25.5%

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

      4. hypot-1-def 25.6%

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

      5. hypot-1-def 25.4%

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

      6. add-sqr-sqrt 25.6%

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

      7. +-commutative 25.6%

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

      8. hypot-1-def 25.4%

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

      9. +-commutative 25.4%

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

      10. div-sub 25.4%

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

   4. Applied egg-rr 25.4%

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

      1. div-sub 25.5%

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

      2. remove-double-neg 25.5%

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

      3. distribute-frac-neg2 25.5%

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

      4. distribute-frac-neg 25.5%

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

      5. neg-sub0 25.5%

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

      6. associate--r- 25.5%

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

      7. neg-sub0 25.5%

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

      8. +-commutative 25.5%

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

      9. fma-undefine 25.5%

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

      10. unpow2 25.5%

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

      11. +-commutative 25.5%

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

      12. associate-+l+ 25.4%

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

      13. sub-neg 25.4%

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

      14. +-inverses 25.4%

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

      15. metadata-eval 25.4%

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

      16. neg-sub0 25.4%

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

      17. associate--r- 25.4%

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

      18. neg-sub0 25.4%

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

      19. +-commutative 25.4%

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

      20. sub-neg 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   9. Simplified 98.2%

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

       1. unpow2 98.9%

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

   11. Applied egg-rr 98.2%

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

   if 1 < x

   1. Initial program 53.2%

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

   3. Taylor expanded in x around 0 44.8%

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

      1. log1p-define 44.8%

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

      2. rem-square-sqrt 44.8%

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

      3. fabs-sqr 44.8%

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

      4. rem-square-sqrt 44.8%

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

   5. Simplified 44.8%

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

Alternative 10: 62.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. hypot-1-def 45.3%

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

      3. flip-+ 19.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

      4. hypot-1-def 19.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      5. hypot-1-def 19.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      6. add-sqr-sqrt 19.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\left(1 + x \cdot x\right)}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      7. +-commutative 19.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\left(x \cdot x + 1\right)}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      8. hypot-1-def 19.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \left(x \cdot x + 1\right)}{\left|x\right| - \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      9. +-commutative 19.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \left(x \cdot x + 1\right)}{\left|x\right| - \sqrt{\color{blue}{x \cdot x + 1}}}\right), x\right) \]

      10. div-sub 19.2%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right|}{\left|x\right| - \sqrt{x \cdot x + 1}} - \frac{x \cdot x + 1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

   4. Applied egg-rr 20.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. div-sub 20.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

      2. remove-double-neg 20.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\left(-\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right)}, x\right) \]

      3. distribute-frac-neg2 20.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(-\color{blue}{\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

      4. distribute-frac-neg 20.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]

      5. neg-sub0 20.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      6. associate--r- 20.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      7. neg-sub0 20.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      8. +-commutative 20.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      9. fma-undefine 20.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(x \cdot x + 1\right)} + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      10. unpow2 20.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left(\color{blue}{{x}^{2}} + 1\right) + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      11. +-commutative 20.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(1 + {x}^{2}\right)} + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      12. associate-+l+ 31.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1 + \left({x}^{2} + \left(-{x}^{2}\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      13. sub-neg 31.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1 + \color{blue}{\left({x}^{2} - {x}^{2}\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      14. +-inverses 45.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1 + \color{blue}{0}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      15. metadata-eval 45.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

      16. neg-sub0 45.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

      17. associate--r- 45.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

      18. neg-sub0 45.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      19. +-commutative 45.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right), x\right) \]

      20. sub-neg 45.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

   6. Simplified 45.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]

   7. Taylor expanded in x around 0 74.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   8. Step-by-step derivation

      1. *-commutative 74.1%

         \[\leadsto \mathsf{copysign}\left(x \cdot \left(1 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right), x\right) \]

   9. Simplified 74.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot -0.16666666666666666\right)}, x\right) \]
   10. Step-by-step derivation

       1. unpow2 74.5%

          \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.075, -0.16666666666666666\right) \cdot {x}^{3}, x\right) \]

   11. Applied egg-rr 74.1%

       \[\leadsto \mathsf{copysign}\left(x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right), x\right) \]

   if 1 < x

   1. Initial program 53.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 44.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define 44.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

      2. rem-square-sqrt 44.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

      3. fabs-sqr 44.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]

      4. rem-square-sqrt 44.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right), x\right) \]

   5. Simplified 44.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 54.2% 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 37.3%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 37.3%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]

   2. hypot-1-def 57.4%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. flip-+ 15.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   4. hypot-1-def 15.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   5. hypot-1-def 15.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   6. add-sqr-sqrt 15.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\left(1 + x \cdot x\right)}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   7. +-commutative 15.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \color{blue}{\left(x \cdot x + 1\right)}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   8. hypot-1-def 15.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \left(x \cdot x + 1\right)}{\left|x\right| - \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

   9. +-commutative 15.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| - \left(x \cdot x + 1\right)}{\left|x\right| - \sqrt{\color{blue}{x \cdot x + 1}}}\right), x\right) \]

   10. div-sub 15.5%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right|}{\left|x\right| - \sqrt{x \cdot x + 1}} - \frac{x \cdot x + 1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

4. Applied egg-rr 16.3%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
5. Step-by-step derivation

   1. div-sub 16.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]

   2. remove-double-neg 16.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\left(-\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)\right)}, x\right) \]

   3. distribute-frac-neg2 16.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(-\color{blue}{\frac{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

   4. distribute-frac-neg 16.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]

   5. neg-sub0 16.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0 - \left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   6. associate--r- 16.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(0 - {x}^{2}\right) + \mathsf{fma}\left(x, x, 1\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   7. neg-sub0 16.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(-{x}^{2}\right)} + \mathsf{fma}\left(x, x, 1\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   8. +-commutative 16.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right) + \left(-{x}^{2}\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   9. fma-undefine 16.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(x \cdot x + 1\right)} + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   10. unpow2 16.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left(\color{blue}{{x}^{2}} + 1\right) + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   11. +-commutative 16.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left(1 + {x}^{2}\right)} + \left(-{x}^{2}\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   12. associate-+l+ 24.9%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1 + \left({x}^{2} + \left(-{x}^{2}\right)\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   13. sub-neg 24.9%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1 + \color{blue}{\left({x}^{2} - {x}^{2}\right)}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   14. +-inverses 35.4%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1 + \color{blue}{0}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   15. metadata-eval 35.4%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{1}}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   16. neg-sub0 36.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]

   17. associate--r- 36.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

   18. neg-sub0 36.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   19. +-commutative 36.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right), x\right) \]

   20. sub-neg 36.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

6. Simplified 36.8%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]

7. Taylor expanded in x around 0 58.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
8. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

float code(float x) {
	float t_0 = 1.0f / fabsf(x);
	return copysignf(log1pf((fabsf(x) + (fabsf(x) / (hypotf(1.0f, t_0) + t_0)))), x);
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) / abs(x))
	return copysign(log1p(Float32(abs(x) + Float32(abs(x) / Float32(hypot(Float32(1.0), t_0) + t_0)))), x)
end

Reproduce

herbie shell --seed 2024141 
(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))