Rust f32::asinh

Percentage Accurate: 37.1% → 99.4%

Time: 9.7s

Alternatives: 9

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: 37.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.20000000298023224)
       (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 (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		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((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(Float32(0.075) + Float32((x ^ Float32(2.0)) * Float32(-0.044642857142857144)))) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.20000000298023224))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.20000000298023224))
		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((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 59.3%

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

      1. flip-+ 6.3%

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

      2. clear-num 6.2%

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

      3. log-div 6.2%

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

      4. metadata-eval 6.2%

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

      5. +-commutative 6.2%

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

      6. hypot-1-def 6.0%

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

      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 5.9%

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

   4. Applied egg-rr 7.4%

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

      1. neg-sub0 7.4%

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

      2. div-sub 7.4%

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

      3. *-rgt-identity 7.4%

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

      4. associate-/l* 7.4%

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

      5. fmm-def 7.4%

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

      6. fma-undefine 7.4%

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

      7. unpow2 7.4%

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

      8. associate--r+ 7.4%

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

      9. +-inverses 7.4%

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

      10. metadata-eval 7.4%

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

      11. metadata-eval 7.4%

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

      12. *-rgt-identity 7.4%

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

      13. associate-/l* 7.4%

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

      14. fma-undefine 7.4%

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

      15. unpow2 7.4%

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

   6. Simplified 99.8%

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

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

   1. Initial program 20.8%

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

      1. +-commutative 20.8%

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

      2. hypot-1-def 20.8%

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

      3. flip-+ 21.0%

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

      4. hypot-1-def 21.0%

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

      5. hypot-1-def 20.9%

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

      6. add-sqr-sqrt 20.9%

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

      7. +-commutative 20.9%

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

      8. hypot-1-def 21.0%

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

      9. +-commutative 21.0%

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

      10. div-sub 21.0%

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

   4. Applied egg-rr 20.8%

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

      1. div-sub 20.8%

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

      2. remove-double-neg 20.8%

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

      3. distribute-frac-neg2 20.8%

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

      4. neg-mul-1 20.8%

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

      5. associate-/r* 20.8%

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

      6. distribute-neg-frac2 20.8%

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

      7. fma-undefine 20.8%

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

      8. unpow2 20.8%

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

      9. associate--r+ 20.8%

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

      10. +-inverses 20.8%

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

      11. metadata-eval 20.8%

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

      12. metadata-eval 20.8%

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

      13. neg-sub0 20.8%

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

      14. associate--r- 20.8%

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

      15. neg-sub0 20.8%

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

      16. +-commutative 20.8%

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

      17. sub-neg 20.8%

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

   6. Simplified 20.8%

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

   7. Taylor expanded in x around 0 100.0%

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

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

   1. Initial program 57.2%

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

      1. +-commutative 57.2%

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

      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. Simplified 100.0%

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

   5. Taylor expanded in x around 0 57.2%

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

      1. rem-square-sqrt 57.2%

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

      2. fabs-sqr 57.2%

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

      3. metadata-eval 57.2%

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

      4. unpow2 57.2%

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

      5. hypot-undefine 100.0%

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

      6. rem-square-sqrt 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.03999999910593033)
       (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
       (copysign (- (log (/ 1.0 (+ x (hypot 1.0 x))))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.03999999910593033f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(-logf((1.0f / (x + hypotf(1.0f, x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.03999999910593033))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / Float32(x + hypot(Float32(1.0), x))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.20000000298023224))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.03999999910593033))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(-log((single(1.0) / (x + hypot(single(1.0), x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 59.3%

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

      1. flip-+ 6.3%

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

      2. clear-num 6.2%

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

      3. log-div 6.2%

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

      4. metadata-eval 6.2%

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

      5. +-commutative 6.2%

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

      6. hypot-1-def 6.0%

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

      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 5.9%

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

   4. Applied egg-rr 7.4%

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

      1. neg-sub0 7.4%

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

      2. div-sub 7.4%

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

      3. *-rgt-identity 7.4%

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

      4. associate-/l* 7.4%

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

      5. fmm-def 7.4%

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

      6. fma-undefine 7.4%

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

      7. unpow2 7.4%

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

      8. associate--r+ 7.4%

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

      9. +-inverses 7.4%

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

      10. metadata-eval 7.4%

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

      11. metadata-eval 7.4%

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

      12. *-rgt-identity 7.4%

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

      13. associate-/l* 7.4%

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

      14. fma-undefine 7.4%

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

      15. unpow2 7.4%

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

   6. Simplified 99.8%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0399999991

   1. Initial program 20.3%

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

      1. +-commutative 20.3%

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

      2. hypot-1-def 20.3%

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

      3. flip-+ 20.4%

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

      4. hypot-1-def 20.4%

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

      5. hypot-1-def 20.4%

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

      6. add-sqr-sqrt 20.4%

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

      7. +-commutative 20.4%

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

      8. hypot-1-def 20.4%

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

      9. +-commutative 20.4%

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

      10. div-sub 20.4%

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

   4. Applied egg-rr 20.3%

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

      1. div-sub 20.3%

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

      2. remove-double-neg 20.3%

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

      3. distribute-frac-neg2 20.3%

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

      4. neg-mul-1 20.3%

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

      5. associate-/r* 20.3%

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

      6. distribute-neg-frac2 20.3%

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

      7. fma-undefine 20.3%

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

      8. unpow2 20.3%

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

      9. associate--r+ 20.3%

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

      10. +-inverses 20.3%

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

      11. metadata-eval 20.3%

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

      12. metadata-eval 20.3%

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

      13. neg-sub0 20.3%

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

      14. associate--r- 20.3%

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

      15. neg-sub0 20.3%

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

      16. +-commutative 20.3%

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

      17. sub-neg 20.3%

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

   6. Simplified 20.3%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. unpow3 100.0%

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

   9. Simplified 100.0%

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

   if 0.0399999991 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 57.8%

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

      1. flip-+ 9.2%

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

      2. clear-num 9.2%

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

      3. log-div 9.2%

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

      4. metadata-eval 9.2%

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

      5. +-commutative 9.2%

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

      6. hypot-1-def 9.2%

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

      7. add-sqr-sqrt 9.4%

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

      8. fabs-sqr 9.4%

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

      9. add-sqr-sqrt 9.2%

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

   4. Applied egg-rr 9.2%

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

      1. neg-sub0 9.2%

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

      2. div-sub 9.2%

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

      3. *-rgt-identity 9.2%

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

      4. associate-/l* 9.2%

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

      5. fmm-def 9.2%

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

      6. fma-undefine 9.2%

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

      7. unpow2 9.2%

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

      8. associate--r+ 9.2%

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

      9. +-inverses 9.2%

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

      10. metadata-eval 9.2%

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

      11. metadata-eval 9.2%

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

      12. *-rgt-identity 9.2%

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

      13. associate-/l* 9.2%

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

      14. fma-undefine 9.2%

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

      15. unpow2 9.2%

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

   6. Simplified 14.9%

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

      1. flip-- 12.5%

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

      2. div-inv 12.5%

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

      3. hypot-undefine 12.5%

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

      4. hypot-undefine 12.5%

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

      5. metadata-eval 12.5%

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

      6. metadata-eval 12.5%

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

      7. add-sqr-sqrt 14.0%

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

      8. pow2 14.0%

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

      9. pow2 14.0%

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

      10. +-commutative 14.0%

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

   8. Applied egg-rr 14.0%

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

      1. associate--l+ 54.9%

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

      2. +-inverses 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

Alternative 3: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.20000000298023224)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.03999999910593033)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.20000000298023224f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.03999999910593033f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.20000000298023224))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.03999999910593033))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.20000000298023224))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.03999999910593033))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.200000003

   1. Initial program 59.3%

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

      1. flip-+ 6.3%

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

      2. clear-num 6.2%

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

      3. log-div 6.2%

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

      4. metadata-eval 6.2%

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

      5. +-commutative 6.2%

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

      6. hypot-1-def 6.0%

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

      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 5.9%

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

   4. Applied egg-rr 7.4%

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

      1. neg-sub0 7.4%

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

      2. div-sub 7.4%

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

      3. *-rgt-identity 7.4%

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

      4. associate-/l* 7.4%

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

      5. fmm-def 7.4%

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

      6. fma-undefine 7.4%

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

      7. unpow2 7.4%

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

      8. associate--r+ 7.4%

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

      9. +-inverses 7.4%

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

      10. metadata-eval 7.4%

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

      11. metadata-eval 7.4%

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

      12. *-rgt-identity 7.4%

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

      13. associate-/l* 7.4%

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

      14. fma-undefine 7.4%

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

      15. unpow2 7.4%

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

   6. Simplified 99.8%

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

   if -0.200000003 < x < 0.0399999991

   1. Initial program 20.3%

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

      1. +-commutative 20.3%

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

      2. hypot-1-def 20.3%

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

      3. flip-+ 20.4%

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

      4. hypot-1-def 20.4%

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

      5. hypot-1-def 20.4%

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

      6. add-sqr-sqrt 20.4%

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

      7. +-commutative 20.4%

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

      8. hypot-1-def 20.4%

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

      9. +-commutative 20.4%

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

      10. div-sub 20.4%

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

   4. Applied egg-rr 20.3%

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

      1. div-sub 20.3%

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

      2. remove-double-neg 20.3%

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

      3. distribute-frac-neg2 20.3%

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

      4. neg-mul-1 20.3%

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

      5. associate-/r* 20.3%

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

      6. distribute-neg-frac2 20.3%

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

      7. fma-undefine 20.3%

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

      8. unpow2 20.3%

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

      9. associate--r+ 20.3%

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

      10. +-inverses 20.3%

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

      11. metadata-eval 20.3%

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

      12. metadata-eval 20.3%

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

      13. neg-sub0 20.3%

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

      14. associate--r- 20.3%

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

      15. neg-sub0 20.3%

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

      16. +-commutative 20.3%

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

      17. sub-neg 20.3%

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

   6. Simplified 20.3%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. unpow3 100.0%

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

   9. Simplified 100.0%

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

   if 0.0399999991 < x

   1. Initial program 57.8%

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

      1. +-commutative 57.8%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. rem-square-sqrt 57.7%

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

      2. fabs-sqr 57.7%

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

      3. metadata-eval 57.7%

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

      4. unpow2 57.7%

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

      5. hypot-undefine 99.9%

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

      6. rem-square-sqrt 99.9%

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

   7. Simplified 99.9%

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

Alternative 4: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.03999999910593033)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.03999999910593033f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.03999999910593033))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.03999999910593033))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 8.2%

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

      9. associate-*r/ 8.2%

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

      10. metadata-eval 8.2%

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

      11. distribute-neg-frac 8.2%

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

      12. metadata-eval 8.2%

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

   5. Simplified 8.2%

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

   6. Taylor expanded in x around 0 99.2%

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

   if -10 < x < 0.0399999991

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

      2. hypot-1-def 21.4%

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

      3. flip-+ 21.5%

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

      4. hypot-1-def 21.6%

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

      5. hypot-1-def 21.5%

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

      6. add-sqr-sqrt 21.6%

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

      7. +-commutative 21.6%

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

      8. hypot-1-def 21.5%

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

      9. +-commutative 21.5%

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

      10. div-sub 21.6%

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

   4. Applied egg-rr 21.4%

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

      1. div-sub 21.4%

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

      2. remove-double-neg 21.4%

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

      3. distribute-frac-neg2 21.4%

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

      4. neg-mul-1 21.4%

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

      5. associate-/r* 21.4%

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

      6. distribute-neg-frac2 21.4%

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

      7. fma-undefine 21.4%

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

      8. unpow2 21.4%

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

      9. associate--r+ 21.4%

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

      10. +-inverses 21.4%

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

      11. metadata-eval 21.4%

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

      12. metadata-eval 21.4%

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

      13. neg-sub0 21.4%

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

      14. associate--r- 21.4%

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

      15. neg-sub0 21.4%

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

      16. +-commutative 21.4%

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

      17. sub-neg 21.4%

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

   6. Simplified 21.4%

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

   7. Taylor expanded in x around 0 99.3%

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

      1. distribute-rgt-in 99.3%

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

      2. *-lft-identity 99.3%

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

      3. associate-*l* 99.3%

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

      4. unpow2 99.3%

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

      5. unpow3 99.3%

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

   9. Simplified 99.3%

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

   if 0.0399999991 < x

   1. Initial program 57.8%

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

      1. +-commutative 57.8%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. rem-square-sqrt 57.7%

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

      2. fabs-sqr 57.7%

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

      3. metadata-eval 57.7%

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

      4. unpow2 57.7%

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

      5. hypot-undefine 99.9%

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

      6. rem-square-sqrt 99.9%

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

   7. Simplified 99.9%

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

Alternative 5: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.20000000298023224)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), 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(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	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 < -10

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 8.2%

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

      9. associate-*r/ 8.2%

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

      10. metadata-eval 8.2%

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

      11. distribute-neg-frac 8.2%

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

      12. metadata-eval 8.2%

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

   5. Simplified 8.2%

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

   6. Taylor expanded in x around 0 99.2%

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

   if -10 < x < 0.200000003

   1. Initial program 22.0%

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

      1. +-commutative 22.0%

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

      2. hypot-1-def 21.9%

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

      3. flip-+ 22.1%

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

      4. hypot-1-def 22.1%

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

      5. hypot-1-def 22.0%

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

      6. add-sqr-sqrt 22.1%

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

      7. +-commutative 22.1%

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

      8. hypot-1-def 22.0%

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

      9. +-commutative 22.0%

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

      10. div-sub 22.1%

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

   4. Applied egg-rr 21.9%

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

      1. div-sub 21.9%

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

      2. remove-double-neg 21.9%

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

      3. distribute-frac-neg2 21.9%

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

      4. neg-mul-1 21.9%

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

      5. associate-/r* 21.9%

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

      6. distribute-neg-frac2 21.9%

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

      7. fma-undefine 21.9%

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

      8. unpow2 21.9%

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

      9. associate--r+ 22.0%

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

      10. +-inverses 22.0%

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

      11. metadata-eval 22.0%

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

      12. metadata-eval 22.0%

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

      13. neg-sub0 22.0%

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

      14. associate--r- 22.0%

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

      15. neg-sub0 22.0%

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

      16. +-commutative 22.0%

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

      17. sub-neg 22.0%

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

   6. Simplified 22.0%

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

   7. Taylor expanded in x around 0 99.1%

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

      1. distribute-rgt-in 99.1%

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

      2. *-lft-identity 99.1%

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

      3. associate-*l* 99.1%

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

      4. unpow2 99.1%

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

      5. unpow3 99.1%

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

   9. Simplified 99.1%

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

   if 0.200000003 < x

   1. Initial program 57.2%

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

   3. Taylor expanded in x around inf 96.9%

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

      1. +-commutative 96.9%

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

      2. rem-square-sqrt 96.9%

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

      3. fabs-sqr 96.9%

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

      4. rem-square-sqrt 96.9%

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

      5. *-inverses 96.9%

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

      6. metadata-eval 96.9%

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

   5. Simplified 96.9%

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

Alternative 6: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.20000000298023224)
     (copysign x x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 8.2%

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

      9. associate-*r/ 8.2%

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

      10. metadata-eval 8.2%

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

      11. distribute-neg-frac 8.2%

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

      12. metadata-eval 8.2%

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

   5. Simplified 8.2%

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

   6. Taylor expanded in x around 0 99.2%

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

   if -10 < x < 0.200000003

   1. Initial program 22.0%

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

   3. Taylor expanded in x around 0 19.1%

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

      1. log1p-define 94.0%

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

      2. rem-square-sqrt 46.9%

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

      3. fabs-sqr 46.9%

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

      4. rem-square-sqrt 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in x around 0 97.8%

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

   if 0.200000003 < x

   1. Initial program 57.2%

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

   3. Taylor expanded in x around inf 96.9%

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

      1. +-commutative 96.9%

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

      2. rem-square-sqrt 96.9%

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

      3. fabs-sqr 96.9%

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

      4. rem-square-sqrt 96.9%

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

      5. *-inverses 96.9%

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

      6. metadata-eval 96.9%

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

   5. Simplified 96.9%

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

Alternative 7: 75.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.20000000298023224) (copysign x x) (copysign (log (* x 2.0)) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.200000003

   1. Initial program 31.8%

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

   3. Taylor expanded in x around 0 26.0%

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

      1. log1p-define 80.4%

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

      2. rem-square-sqrt 34.0%

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

      3. fabs-sqr 34.0%

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

      4. rem-square-sqrt 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in x around 0 74.0%

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

   if 0.200000003 < x

   1. Initial program 57.2%

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

   3. Taylor expanded in x around inf 96.9%

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

      1. +-commutative 96.9%

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

      2. rem-square-sqrt 96.9%

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

      3. fabs-sqr 96.9%

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

      4. rem-square-sqrt 96.9%

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

      5. *-inverses 96.9%

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

      6. metadata-eval 96.9%

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

   5. Simplified 96.9%

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

Alternative 8: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.20000000298023224) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.200000003

   1. Initial program 31.8%

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

   3. Taylor expanded in x around 0 26.0%

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

      1. log1p-define 80.4%

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

      2. rem-square-sqrt 34.0%

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

      3. fabs-sqr 34.0%

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

      4. rem-square-sqrt 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in x around 0 74.0%

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

   if 0.200000003 < x

   1. Initial program 57.2%

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

   3. Taylor expanded in x around 0 43.9%

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

      1. log1p-define 43.9%

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

      2. rem-square-sqrt 43.9%

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

      3. fabs-sqr 43.9%

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

      4. rem-square-sqrt 43.9%

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

   5. Simplified 43.9%

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

Alternative 9: 54.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in x around 0 30.9%

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

   1. log1p-define 70.4%

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

   2. rem-square-sqrt 36.7%

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

   3. fabs-sqr 36.7%

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

   4. rem-square-sqrt 61.5%

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

5. Simplified 61.5%

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

6. Taylor expanded in x around 0 57.0%

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

Developer Target 1: 99.4% 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 2024139 
(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))