Rust f32::asinh

Percentage Accurate: 38.4% → 99.4%

Time: 10.0s

Alternatives: 12

Speedup: 4.0×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (asinh x))

C

float code(float x) {
	return asinhf(x);
}

Julia

function code(x)
	return asinh(x)
end

MATLAB

function tmp = code(x)
	tmp = asinh(x);
end

Initial Program: 38.4% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -5.0)
   (copysign (log (/ -0.5 x)) x)
   (copysign (log1p (+ x (* x (/ x (+ 1.0 (hypot 1.0 x)))))) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf(log1pf((x + (x * (x / (1.0f + hypotf(1.0f, x)))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x) <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(x * Float32(x / Float32(Float32(1.0) + hypot(Float32(1.0), x)))))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.1%

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

   3. Taylor expanded in x around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. *-commutative 97.9%

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

      3. distribute-rgt-neg-in 97.9%

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

      4. mul-1-neg 97.9%

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

      5. unsub-neg 97.9%

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

      6. sub-neg 97.9%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 9.4%

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

      10. associate-*r/ 9.4%

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

      11. metadata-eval 9.4%

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

      12. distribute-neg-frac 9.4%

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

      13. metadata-eval 9.4%

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

   5. Simplified 9.4%

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

   6. Taylor expanded in x around 0 99.9%

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

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

   1. Initial program 40.4%

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

      1. add-log-exp 22.0%

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

      2. *-un-lft-identity 22.0%

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

      3. log-prod 22.0%

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

      4. metadata-eval 22.0%

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

      5. add-log-exp 40.4%

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

      6. add-sqr-sqrt 31.2%

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

      7. fabs-sqr 31.2%

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

      8. add-sqr-sqrt 40.3%

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

   4. Applied egg-rr 40.3%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(0 + x\right)} + \sqrt{x \cdot x + 1}\right), x\right) \]
   5. Step-by-step derivation

      1. +-lft-identity 40.3%

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

      2. +-commutative 40.3%

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

      3. hypot-1-def 56.6%

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

      4. pow1 56.6%

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

      5. metadata-eval 56.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{\color{blue}{\left(6 \cdot 0.16666666666666666\right)}} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      6. pow-pow 26.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{\left({x}^{6}\right)}^{0.16666666666666666}} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      7. log1p-expm1-u 26.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left({\left({x}^{6}\right)}^{0.16666666666666666} + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      8. expm1-undefine 26.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left({\left({x}^{6}\right)}^{0.16666666666666666} + \mathsf{hypot}\left(1, x\right)\right)} - 1}\right), x\right) \]

      9. pow-pow 56.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(e^{\log \left(\color{blue}{{x}^{\left(6 \cdot 0.16666666666666666\right)}} + \mathsf{hypot}\left(1, x\right)\right)} - 1\right), x\right) \]

      10. metadata-eval 56.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(e^{\log \left({x}^{\color{blue}{1}} + \mathsf{hypot}\left(1, x\right)\right)} - 1\right), x\right) \]

      11. pow1 56.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(e^{\log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)} - 1\right), x\right) \]

      12. add-exp-log 56.5%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(x + \mathsf{hypot}\left(1, x\right)\right)} - 1\right), x\right) \]

   6. Applied egg-rr 56.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)}, x\right) \]
   7. Step-by-step derivation

      1. associate--l+ 97.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)}\right), x\right) \]

   8. Simplified 97.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. flip-- 81.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - 1 \cdot 1}{\mathsf{hypot}\left(1, x\right) + 1}}\right), x\right) \]

      2. div-inv 81.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}}\right), x\right) \]

      3. metadata-eval 81.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - \color{blue}{1}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]

      4. sub-neg 81.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) + \left(-1\right)\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]

      5. hypot-undefine 81.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right) + \left(-1\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]

      6. hypot-undefine 80.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} + \left(-1\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]

      7. metadata-eval 80.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{\color{blue}{1} + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} + \left(-1\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]

      8. metadata-eval 80.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{1 + x \cdot x} \cdot \sqrt{\color{blue}{1} + x \cdot x} + \left(-1\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]

      9. add-sqr-sqrt 81.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\left(1 + x \cdot x\right)} + \left(-1\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]

      10. pow2 81.3%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\left(1 + \color{blue}{{x}^{2}}\right) + \left(-1\right)\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]

      11. metadata-eval 81.3%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\left(1 + {x}^{2}\right) + \color{blue}{-1}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]

      12. +-commutative 81.3%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\left(1 + {x}^{2}\right) + -1\right) \cdot \frac{1}{\color{blue}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

   10. Applied egg-rr 81.3%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\left(\left(1 + {x}^{2}\right) + -1\right) \cdot \frac{1}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 81.3%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{\left(\left(1 + {x}^{2}\right) + -1\right) \cdot 1}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

       2. *-rgt-identity 81.3%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{\left(1 + {x}^{2}\right) + -1}}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

       3. +-commutative 81.3%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{\left({x}^{2} + 1\right)} + -1}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

       4. associate-+l+ 83.5%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{{x}^{2} + \left(1 + -1\right)}}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

       5. metadata-eval 83.5%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{{x}^{2} + \color{blue}{0}}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   12. Simplified 83.5%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{{x}^{2} + 0}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
   13. Step-by-step derivation

       1. +-rgt-identity 83.5%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{{x}^{2}}}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

       2. unpow2 83.5%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{x \cdot x}}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

       3. associate-/l* 99.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{x \cdot \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

   14. Applied egg-rr 99.6%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{x \cdot \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.20000000298023224)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 0.019999999552965164)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.20000000298023224f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (x <= 0.019999999552965164f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.20000000298023224))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.019999999552965164))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.20000000298023224))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (x <= single(0.019999999552965164))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.200000003

   1. Initial program 64.4%

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

      1. +-commutative 64.4%

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

      2. hypot-1-def 98.0%

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

      3. flip-+ 13.0%

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

      4. hypot-1-def 12.7%

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

      5. hypot-1-def 11.1%

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

      6. add-sqr-sqrt 12.7%

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

      7. +-commutative 12.7%

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

      8. hypot-1-def 11.2%

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

      9. +-commutative 11.2%

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

      10. div-sub 11.2%

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

   4. Applied egg-rr 15.8%

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

      1. div-sub 19.5%

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

      2. remove-double-neg 19.5%

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

      3. distribute-neg-frac2 19.5%

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

      4. distribute-frac-neg 19.5%

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

      5. neg-sub0 19.5%

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

      6. associate--r- 19.5%

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

      7. neg-sub0 19.5%

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

      8. +-commutative 19.5%

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

      9. fma-undefine 19.5%

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

      10. unpow2 19.5%

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

      11. +-commutative 19.5%

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

      12. associate-+l+ 62.1%

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

      13. sub-neg 62.1%

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

      14. +-inverses 98.1%

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

      15. metadata-eval 98.1%

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

      16. neg-sub0 98.1%

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

      17. associate--r- 98.1%

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

      18. neg-sub0 98.1%

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

      19. +-commutative 98.1%

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

      20. sub-neg 98.1%

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

   6. Simplified 98.1%

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

   if -0.200000003 < x < 0.0199999996

   1. Initial program 23.8%

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

      1. add-log-exp 23.8%

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

      2. *-un-lft-identity 23.8%

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

      3. log-prod 23.8%

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

      4. metadata-eval 23.8%

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

      5. add-log-exp 23.8%

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

      6. add-sqr-sqrt 12.7%

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

      7. fabs-sqr 12.7%

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

      8. add-sqr-sqrt 24.2%

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

   4. Applied egg-rr 24.2%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. unpow2 100.0%

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

      6. cube-mult 100.0%

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

   7. Simplified 100.0%

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

   if 0.0199999996 < x

   1. Initial program 59.7%

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

   3. Taylor expanded in x around 0 59.7%

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

      1. rem-square-sqrt 59.7%

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

      2. fabs-sqr 59.7%

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

      3. metadata-eval 59.7%

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

      4. unpow2 59.7%

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

      5. hypot-undefine 99.7%

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

      6. rem-square-sqrt 99.8%

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

   5. Simplified 99.8%

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

Alternative 3: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.20000000298023224)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (if (<= x 0.019999999552965164)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.20000000298023224f) {
		tmp = copysignf(logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.019999999552965164f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.20000000298023224))
		tmp = copysign(log(Float32(hypot(Float32(1.0), x) - x)), x);
	elseif (x <= Float32(0.019999999552965164))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.20000000298023224))
		tmp = sign(x) * abs(log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.019999999552965164))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.200000003

   1. Initial program 64.4%

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

      1. +-commutative 64.4%

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

      2. hypot-1-def 98.0%

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

   3. Simplified 98.0%

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

      1. add-sqr-sqrt -0.0%

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

      2. fabs-sqr -0.0%

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

      3. sqrt-prod 64.5%

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

      4. pow1/2 64.5%

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

      5. add-cbrt-cube 27.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\color{blue}{\left(\sqrt[3]{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)}}^{0.5} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      6. pow1/3 27.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\color{blue}{\left({\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}^{0.3333333333333333}\right)}}^{0.5} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      7. pow-pow 27.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}^{\left(0.3333333333333333 \cdot 0.5\right)}} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      8. pow3 27.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\color{blue}{\left({\left(x \cdot x\right)}^{3}\right)}}^{\left(0.3333333333333333 \cdot 0.5\right)} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      9. pow2 27.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left({\color{blue}{\left({x}^{2}\right)}}^{3}\right)}^{\left(0.3333333333333333 \cdot 0.5\right)} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      10. pow-pow 27.5%

          \[\leadsto \mathsf{copysign}\left(\log \left({\color{blue}{\left({x}^{\left(2 \cdot 3\right)}\right)}}^{\left(0.3333333333333333 \cdot 0.5\right)} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      11. metadata-eval 27.5%

          \[\leadsto \mathsf{copysign}\left(\log \left({\left({x}^{\color{blue}{6}}\right)}^{\left(0.3333333333333333 \cdot 0.5\right)} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      12. metadata-eval 27.5%

          \[\leadsto \mathsf{copysign}\left(\log \left({\left({x}^{6}\right)}^{\color{blue}{0.16666666666666666}} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

   6. Applied egg-rr 27.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{\left({x}^{6}\right)}^{0.16666666666666666}} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

   7. Taylor expanded in x around -inf 64.4%

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

      1. metadata-eval 64.4%

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

      2. unpow2 64.4%

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

      3. hypot-undefine 98.0%

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

      4. metadata-eval 98.0%

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

      5. pow-base-1 98.0%

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

      6. *-lft-identity 98.0%

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

      7. cancel-sign-sub-inv 98.0%

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

      8. *-lft-identity 98.0%

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

   9. Simplified 98.0%

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

   if -0.200000003 < x < 0.0199999996

   1. Initial program 23.8%

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

      1. add-log-exp 23.8%

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

      2. *-un-lft-identity 23.8%

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

      3. log-prod 23.8%

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

      4. metadata-eval 23.8%

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

      5. add-log-exp 23.8%

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

      6. add-sqr-sqrt 12.7%

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

      7. fabs-sqr 12.7%

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

      8. add-sqr-sqrt 24.2%

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

   4. Applied egg-rr 24.2%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. unpow2 100.0%

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

      6. cube-mult 100.0%

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

   7. Simplified 100.0%

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

   if 0.0199999996 < x

   1. Initial program 59.7%

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

   3. Taylor expanded in x around 0 59.7%

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

      1. rem-square-sqrt 59.7%

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

      2. fabs-sqr 59.7%

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

      3. metadata-eval 59.7%

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

      4. unpow2 59.7%

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

      5. hypot-undefine 99.7%

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

      6. rem-square-sqrt 99.8%

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

   5. Simplified 99.8%

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

Alternative 4: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.019999999552965164)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.019999999552965164f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.019999999552965164))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.019999999552965164))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 63.8%

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

   3. Taylor expanded in x around -inf 95.8%

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

      1. mul-1-neg 95.8%

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

      2. *-commutative 95.8%

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

      3. distribute-rgt-neg-in 95.8%

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

      4. mul-1-neg 95.8%

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

      5. unsub-neg 95.8%

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

      6. sub-neg 95.8%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 13.7%

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

      10. associate-*r/ 13.7%

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

      11. metadata-eval 13.7%

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

      12. distribute-neg-frac 13.7%

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

      13. metadata-eval 13.7%

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

   5. Simplified 13.7%

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

   6. Taylor expanded in x around 0 95.5%

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

   if -1 < x < 0.0199999996

   1. Initial program 24.4%

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

      1. add-log-exp 24.4%

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

      2. *-un-lft-identity 24.4%

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

      3. log-prod 24.4%

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

      4. metadata-eval 24.4%

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

      5. add-log-exp 24.4%

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

      6. add-sqr-sqrt 12.6%

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

      7. fabs-sqr 12.6%

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

      8. add-sqr-sqrt 24.8%

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

   4. Applied egg-rr 24.8%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. unpow2 99.7%

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

      6. cube-mult 99.7%

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

   7. Simplified 99.7%

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

   if 0.0199999996 < x

   1. Initial program 59.7%

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

   3. Taylor expanded in x around 0 59.7%

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

      1. rem-square-sqrt 59.7%

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

      2. fabs-sqr 59.7%

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

      3. metadata-eval 59.7%

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

      4. unpow2 59.7%

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

      5. hypot-undefine 99.7%

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

      6. rem-square-sqrt 99.8%

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

   5. Simplified 99.8%

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

Alternative 5: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -500.0)
   (copysign (log (/ -0.5 x)) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

float code(float x) {
	float tmp;
	if (x <= -500.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf(log1pf((x + (hypotf(1.0f, x) + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-500.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(hypot(Float32(1.0), x) + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -500

   1. Initial program 60.1%

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

   3. Taylor expanded in x around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. *-commutative 97.9%

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

      3. distribute-rgt-neg-in 97.9%

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

      4. mul-1-neg 97.9%

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

      5. unsub-neg 97.9%

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

      6. sub-neg 97.9%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 9.4%

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

      10. associate-*r/ 9.4%

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

      11. metadata-eval 9.4%

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

      12. distribute-neg-frac 9.4%

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

      13. metadata-eval 9.4%

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

   5. Simplified 9.4%

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

   6. Taylor expanded in x around 0 99.9%

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

   if -500 < x

   1. Initial program 40.4%

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

      1. add-log-exp 22.0%

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

      2. *-un-lft-identity 22.0%

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

      3. log-prod 22.0%

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

      4. metadata-eval 22.0%

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

      5. add-log-exp 40.4%

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

      6. add-sqr-sqrt 31.2%

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

      7. fabs-sqr 31.2%

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

      8. add-sqr-sqrt 40.3%

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

   4. Applied egg-rr 40.3%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(0 + x\right)} + \sqrt{x \cdot x + 1}\right), x\right) \]
   5. Step-by-step derivation

      1. +-lft-identity 40.3%

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

      2. +-commutative 40.3%

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

      3. hypot-1-def 56.6%

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

      4. pow1 56.6%

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

      5. metadata-eval 56.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({x}^{\color{blue}{\left(6 \cdot 0.16666666666666666\right)}} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      6. pow-pow 26.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{\left({x}^{6}\right)}^{0.16666666666666666}} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      7. log1p-expm1-u 26.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left({\left({x}^{6}\right)}^{0.16666666666666666} + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      8. expm1-undefine 26.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left({\left({x}^{6}\right)}^{0.16666666666666666} + \mathsf{hypot}\left(1, x\right)\right)} - 1}\right), x\right) \]

      9. pow-pow 56.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(e^{\log \left(\color{blue}{{x}^{\left(6 \cdot 0.16666666666666666\right)}} + \mathsf{hypot}\left(1, x\right)\right)} - 1\right), x\right) \]

      10. metadata-eval 56.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(e^{\log \left({x}^{\color{blue}{1}} + \mathsf{hypot}\left(1, x\right)\right)} - 1\right), x\right) \]

      11. pow1 56.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(e^{\log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)} - 1\right), x\right) \]

      12. add-exp-log 56.5%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(x + \mathsf{hypot}\left(1, x\right)\right)} - 1\right), x\right) \]

   6. Applied egg-rr 56.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)}, x\right) \]
   7. Step-by-step derivation

      1. associate--l+ 97.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)}\right), x\right) \]

   8. Simplified 97.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 63.8%

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

   3. Taylor expanded in x around -inf 95.8%

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

      1. mul-1-neg 95.8%

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

      2. *-commutative 95.8%

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

      3. distribute-rgt-neg-in 95.8%

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

      4. mul-1-neg 95.8%

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

      5. unsub-neg 95.8%

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

      6. sub-neg 95.8%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 13.7%

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

      10. associate-*r/ 13.7%

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

      11. metadata-eval 13.7%

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

      12. distribute-neg-frac 13.7%

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

      13. metadata-eval 13.7%

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

   5. Simplified 13.7%

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

   6. Taylor expanded in x around 0 95.5%

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

   if -1 < x < 0.5

   1. Initial program 26.1%

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

      1. add-log-exp 26.0%

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

      2. *-un-lft-identity 26.0%

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

      3. log-prod 26.0%

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

      4. metadata-eval 26.0%

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

      5. add-log-exp 26.1%

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

      6. add-sqr-sqrt 14.6%

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

      7. fabs-sqr 14.6%

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

      8. add-sqr-sqrt 26.5%

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

   4. Applied egg-rr 26.5%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. distribute-lft-in 99.1%

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

      2. *-rgt-identity 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*r* 99.1%

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

      5. unpow2 99.1%

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

      6. cube-mult 99.1%

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

   7. Simplified 99.1%

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

   if 0.5 < x

   1. Initial program 58.4%

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

   3. Taylor expanded in x around -inf 5.6%

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

      1. mul-1-neg 5.6%

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

      2. *-commutative 5.6%

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

      3. distribute-rgt-neg-in 5.6%

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

      4. mul-1-neg 5.6%

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

      5. unsub-neg 5.6%

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

      6. sub-neg 5.6%

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

      7. rem-square-sqrt 8.1%

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

      8. fabs-sqr 8.1%

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

      9. rem-square-sqrt 5.6%

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

      10. associate-*r/ 5.6%

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

      11. metadata-eval 5.6%

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

      12. distribute-neg-frac 5.6%

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

      13. metadata-eval 5.6%

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

   5. Simplified 5.6%

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

      1. metadata-eval 5.6%

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

      2. add-sqr-sqrt 5.6%

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

      3. sqrt-prod 5.6%

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

      4. sqr-neg 5.6%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 11.9%

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

      7. frac-2neg 11.9%

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

      8. div-inv 11.9%

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

      9. metadata-eval 11.9%

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

      10. cancel-sign-sub-inv 11.9%

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

      11. div-inv 11.9%

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

   7. Applied egg-rr 11.9%

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

   8. Taylor expanded in x around 0 97.3%

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

Alternative 7: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 63.8%

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

   3. Taylor expanded in x around -inf 95.8%

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

      1. mul-1-neg 95.8%

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

      2. *-commutative 95.8%

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

      3. distribute-rgt-neg-in 95.8%

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

      4. mul-1-neg 95.8%

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

      5. unsub-neg 95.8%

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

      6. sub-neg 95.8%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 13.7%

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

      10. associate-*r/ 13.7%

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

      11. metadata-eval 13.7%

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

      12. distribute-neg-frac 13.7%

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

      13. metadata-eval 13.7%

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

   5. Simplified 13.7%

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

   6. Taylor expanded in x around 0 95.5%

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

   if -1 < x < 0.5

   1. Initial program 26.1%

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

      1. add-log-exp 26.0%

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

      2. *-un-lft-identity 26.0%

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

      3. log-prod 26.0%

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

      4. metadata-eval 26.0%

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

      5. add-log-exp 26.1%

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

      6. add-sqr-sqrt 14.6%

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

      7. fabs-sqr 14.6%

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

      8. add-sqr-sqrt 26.5%

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

   4. Applied egg-rr 26.5%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. unpow2 99.1%

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

   9. Applied egg-rr 99.1%

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

   if 0.5 < x

   1. Initial program 58.4%

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

   3. Taylor expanded in x around -inf 5.6%

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

      1. mul-1-neg 5.6%

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

      2. *-commutative 5.6%

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

      3. distribute-rgt-neg-in 5.6%

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

      4. mul-1-neg 5.6%

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

      5. unsub-neg 5.6%

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

      6. sub-neg 5.6%

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

      7. rem-square-sqrt 8.1%

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

      8. fabs-sqr 8.1%

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

      9. rem-square-sqrt 5.6%

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

      10. associate-*r/ 5.6%

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

      11. metadata-eval 5.6%

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

      12. distribute-neg-frac 5.6%

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

      13. metadata-eval 5.6%

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

   5. Simplified 5.6%

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

      1. metadata-eval 5.6%

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

      2. add-sqr-sqrt 5.6%

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

      3. sqrt-prod 5.6%

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

      4. sqr-neg 5.6%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 11.9%

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

      7. frac-2neg 11.9%

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

      8. div-inv 11.9%

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

      9. metadata-eval 11.9%

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

      10. cancel-sign-sub-inv 11.9%

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

      11. div-inv 11.9%

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

   7. Applied egg-rr 11.9%

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

   8. Taylor expanded in x around 0 97.3%

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

Alternative 8: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 63.8%

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

   3. Taylor expanded in x around -inf 95.8%

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

      1. mul-1-neg 95.8%

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

      2. *-commutative 95.8%

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

      3. distribute-rgt-neg-in 95.8%

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

      4. mul-1-neg 95.8%

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

      5. unsub-neg 95.8%

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

      6. sub-neg 95.8%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 13.7%

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

      10. associate-*r/ 13.7%

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

      11. metadata-eval 13.7%

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

      12. distribute-neg-frac 13.7%

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

      13. metadata-eval 13.7%

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

   5. Simplified 13.7%

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

   6. Taylor expanded in x around 0 95.5%

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

   if -1 < x < 0.5

   1. Initial program 26.1%

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

      1. add-log-exp 26.0%

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

      2. *-un-lft-identity 26.0%

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

      3. log-prod 26.0%

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

      4. metadata-eval 26.0%

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

      5. add-log-exp 26.1%

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

      6. add-sqr-sqrt 14.6%

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

      7. fabs-sqr 14.6%

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

      8. add-sqr-sqrt 26.5%

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

   4. Applied egg-rr 26.5%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. unpow2 99.1%

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

   9. Applied egg-rr 99.1%

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

   if 0.5 < x

   1. Initial program 58.4%

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

   3. Taylor expanded in x around inf 97.3%

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

      1. +-commutative 97.3%

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

      2. rem-square-sqrt 97.3%

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

      3. fabs-sqr 97.3%

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

      4. rem-square-sqrt 97.3%

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

      5. *-inverses 97.3%

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

      6. metadata-eval 97.3%

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

   5. Simplified 97.3%

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

Alternative 9: 84.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 63.1%

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

   3. Taylor expanded in x around -inf 43.8%

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

      1. neg-mul-1 43.8%

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

   5. Simplified 43.8%

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

   if -5 < x < 0.5

   1. Initial program 26.7%

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

      1. add-log-exp 26.6%

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

      2. *-un-lft-identity 26.6%

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

      3. log-prod 26.6%

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

      4. metadata-eval 26.6%

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

      5. add-log-exp 26.7%

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

      6. add-sqr-sqrt 14.5%

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

      7. fabs-sqr 14.5%

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

      8. add-sqr-sqrt 27.1%

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

   4. Applied egg-rr 27.1%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

      1. unpow2 98.6%

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

   9. Applied egg-rr 98.6%

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

   if 0.5 < x

   1. Initial program 58.4%

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

   3. Taylor expanded in x around inf 97.3%

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

      1. +-commutative 97.3%

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

      2. rem-square-sqrt 97.3%

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

      3. fabs-sqr 97.3%

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

      4. rem-square-sqrt 97.3%

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

      5. *-inverses 97.3%

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

      6. metadata-eval 97.3%

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

   5. Simplified 97.3%

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

Alternative 10: 71.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (- x)) x)
   (if (<= x 0.5)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log1p x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 63.1%

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

   3. Taylor expanded in x around -inf 43.8%

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

      1. neg-mul-1 43.8%

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

   5. Simplified 43.8%

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

   if -5 < x < 0.5

   1. Initial program 26.7%

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

      1. add-log-exp 26.6%

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

      2. *-un-lft-identity 26.6%

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

      3. log-prod 26.6%

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

      4. metadata-eval 26.6%

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

      5. add-log-exp 26.7%

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

      6. add-sqr-sqrt 14.5%

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

      7. fabs-sqr 14.5%

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

      8. add-sqr-sqrt 27.1%

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

   4. Applied egg-rr 27.1%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

      1. unpow2 98.6%

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

   9. Applied egg-rr 98.6%

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

   if 0.5 < x

   1. Initial program 58.4%

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

   3. Taylor expanded in x around 0 43.9%

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

      1. log1p-define 43.9%

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

      2. rem-square-sqrt 43.9%

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

      3. fabs-sqr 43.9%

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

      4. rem-square-sqrt 43.9%

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

   5. Simplified 43.9%

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

Alternative 11: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.5)
   (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
   (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.5f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.5))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 37.3%

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

      1. add-log-exp 22.9%

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

      2. *-un-lft-identity 22.9%

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

      3. log-prod 22.9%

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

      4. metadata-eval 22.9%

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

      5. add-log-exp 37.3%

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

      6. add-sqr-sqrt 10.2%

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

      7. fabs-sqr 10.2%

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

      8. add-sqr-sqrt 23.1%

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

   4. Applied egg-rr 23.1%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. *-commutative 72.3%

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

   7. Simplified 72.3%

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

      1. unpow2 72.3%

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

   9. Applied egg-rr 72.3%

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

   if 0.5 < x

   1. Initial program 58.4%

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

   3. Taylor expanded in x around 0 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 12: 53.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 44.0%

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

   1. add-log-exp 19.1%

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

   2. *-un-lft-identity 19.1%

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

   3. log-prod 19.1%

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

   4. metadata-eval 19.1%

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

   5. add-log-exp 44.0%

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

   6. add-sqr-sqrt 25.5%

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

   7. fabs-sqr 25.5%

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

   8. add-sqr-sqrt 34.3%

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

4. Applied egg-rr 34.3%

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

5. Taylor expanded in x around 0 52.9%

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

Developer Target 1: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024177 
(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))