Rust f32::asinh

Percentage Accurate: 37.8% → 98.5%

Time: 9.9s

Alternatives: 8

Speedup: 3.7×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(\log 2 + \frac{0.25}{{x}^{2}}\right) + \log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.019999999552965164)
       (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
       (copysign (+ (+ (log 2.0) (/ 0.25 (pow x 2.0))) (log x)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (t_0 <= 0.019999999552965164f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(((logf(2.0f) + (0.25f / powf(x, 2.0f))) + logf(x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.019999999552965164))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(Float32(Float32(log(Float32(2.0)) + Float32(Float32(0.25) / (x ^ Float32(2.0)))) + log(x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.20000000298023224))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (t_0 <= single(0.019999999552965164))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * single(-0.16666666666666666)))));
	else
		tmp = sign(x) * abs(((log(single(2.0)) + (single(0.25) / (x ^ single(2.0)))) + log(x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.5%

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

   3. Taylor expanded in x around 0 49.5%

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

      1. rem-square-sqrt -0.0%

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

      2. fabs-sqr -0.0%

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

      3. metadata-eval -0.0%

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

      4. unpow2 -0.0%

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

      5. hypot-undefine -0.0%

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

      6. rem-square-sqrt 12.1%

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

   5. Simplified 12.1%

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

      1. +-commutative 12.1%

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

      2. flip-+ 8.8%

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

      3. hypot-undefine 10.0%

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

      4. metadata-eval 10.0%

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

      5. unpow2 10.0%

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

      6. hypot-undefine 8.3%

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

      7. metadata-eval 8.3%

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

      8. unpow2 8.3%

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

      9. add-sqr-sqrt 10.3%

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

      10. +-commutative 10.3%

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

      11. unpow2 10.3%

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

   7. Applied egg-rr 10.3%

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

      1. sub-neg 10.3%

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

      2. +-commutative 10.3%

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

      3. associate-+l+ 46.0%

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

      4. sub-neg 46.0%

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

      5. +-inverses 100.0%

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

      6. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0199999996

   1. Initial program 17.5%

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

   3. Taylor expanded in x around 0 18.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative 18.5%

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

      2. fma-define 18.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. +-commutative 18.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\left|x\right| + 1}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. rem-square-sqrt 10.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. fabs-sqr 10.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. rem-square-sqrt 18.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{x} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      7. log1p-define 99.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      8. rem-square-sqrt 53.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      9. fabs-sqr 53.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      10. rem-square-sqrt 99.9%

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   5. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. unpow2 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 0.0199999996 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 55.3%

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

   3. Taylor expanded in x around 0 55.3%

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

      1. rem-square-sqrt 55.3%

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

      2. fabs-sqr 55.3%

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

      3. metadata-eval 55.3%

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

      4. unpow2 55.3%

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

      5. hypot-undefine 98.6%

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

      6. rem-square-sqrt 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around inf 99.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)}, x\right) \]
   7. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \mathsf{copysign}\left(\log 2 + \color{blue}{\left(0.25 \cdot \frac{1}{{x}^{2}} + -1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]

      2. mul-1-neg 99.1%

         \[\leadsto \mathsf{copysign}\left(\log 2 + \left(0.25 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}\right), x\right) \]

      3. log-rec 99.1%

         \[\leadsto \mathsf{copysign}\left(\log 2 + \left(0.25 \cdot \frac{1}{{x}^{2}} + \left(-\color{blue}{\left(-\log x\right)}\right)\right), x\right) \]

      4. remove-double-neg 99.1%

         \[\leadsto \mathsf{copysign}\left(\log 2 + \left(0.25 \cdot \frac{1}{{x}^{2}} + \color{blue}{\log x}\right), x\right) \]

      5. associate-+r+ 99.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 2 + 0.25 \cdot \frac{1}{{x}^{2}}\right) + \log x}, x\right) \]

      6. associate-*r/ 99.2%

         \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right) + \log x, x\right) \]

      7. metadata-eval 99.2%

         \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \frac{\color{blue}{0.25}}{{x}^{2}}\right) + \log x, x\right) \]

   8. Simplified 99.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 2 + \frac{0.25}{{x}^{2}}\right) + \log x}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.009999999776482582)
       (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
       (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (t_0 <= 0.009999999776482582f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.009999999776482582))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.20000000298023224))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (t_0 <= single(0.009999999776482582))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * single(-0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((abs(x) + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.5%

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

   3. Taylor expanded in x around 0 49.5%

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

      1. rem-square-sqrt -0.0%

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

      2. fabs-sqr -0.0%

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

      3. metadata-eval -0.0%

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

      4. unpow2 -0.0%

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

      5. hypot-undefine -0.0%

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

      6. rem-square-sqrt 12.1%

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

   5. Simplified 12.1%

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

      1. +-commutative 12.1%

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

      2. flip-+ 8.8%

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

      3. hypot-undefine 10.0%

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

      4. metadata-eval 10.0%

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

      5. unpow2 10.0%

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

      6. hypot-undefine 8.3%

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

      7. metadata-eval 8.3%

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

      8. unpow2 8.3%

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

      9. add-sqr-sqrt 10.3%

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

      10. +-commutative 10.3%

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

      11. unpow2 10.3%

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

   7. Applied egg-rr 10.3%

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

      1. sub-neg 10.3%

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

      2. +-commutative 10.3%

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

      3. associate-+l+ 46.0%

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

      4. sub-neg 46.0%

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

      5. +-inverses 100.0%

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

      6. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.00999999978

   1. Initial program 16.9%

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

   3. Taylor expanded in x around 0 18.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative 18.0%

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

      2. fma-define 18.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. +-commutative 18.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\left|x\right| + 1}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. rem-square-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. fabs-sqr 9.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. rem-square-sqrt 17.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{x} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      7. log1p-define 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      8. rem-square-sqrt 52.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      9. fabs-sqr 52.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      10. rem-square-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. unpow2 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 0.00999999978 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 55.9%

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

      1. +-commutative 55.9%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.20000000298023224)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 0.009999999776482582)
     (copysign (* x (+ 1.0 (* (* x x) -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 <= 0.009999999776482582f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -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(0.009999999776482582))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * 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(0.009999999776482582))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * 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 49.5%

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

   3. Taylor expanded in x around 0 49.5%

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

      1. rem-square-sqrt -0.0%

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

      2. fabs-sqr -0.0%

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

      3. metadata-eval -0.0%

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

      4. unpow2 -0.0%

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

      5. hypot-undefine -0.0%

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

      6. rem-square-sqrt 12.1%

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

   5. Simplified 12.1%

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

      1. +-commutative 12.1%

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

      2. flip-+ 8.8%

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

      3. hypot-undefine 10.0%

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

      4. metadata-eval 10.0%

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

      5. unpow2 10.0%

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

      6. hypot-undefine 8.3%

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

      7. metadata-eval 8.3%

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

      8. unpow2 8.3%

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

      9. add-sqr-sqrt 10.3%

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

      10. +-commutative 10.3%

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

      11. unpow2 10.3%

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

   7. Applied egg-rr 10.3%

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

      1. sub-neg 10.3%

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

      2. +-commutative 10.3%

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

      3. associate-+l+ 46.0%

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

      4. sub-neg 46.0%

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

      5. +-inverses 100.0%

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

      6. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   if -0.200000003 < x < 0.00999999978

   1. Initial program 16.9%

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

   3. Taylor expanded in x around 0 18.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative 18.0%

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

      2. fma-define 18.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. +-commutative 18.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\left|x\right| + 1}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. rem-square-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. fabs-sqr 9.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. rem-square-sqrt 17.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{x} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      7. log1p-define 99.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      8. rem-square-sqrt 52.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      9. fabs-sqr 52.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      10. rem-square-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. unpow2 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 0.00999999978 < x

   1. Initial program 55.9%

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

   3. Taylor expanded in x around 0 55.9%

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

      1. rem-square-sqrt 55.9%

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

      2. fabs-sqr 55.9%

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

      3. metadata-eval 55.9%

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

      4. unpow2 55.9%

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

      5. hypot-undefine 98.5%

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

      6. rem-square-sqrt 98.5%

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

   5. Simplified 98.5%

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

Alternative 4: 83.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-e}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ (- E) x)) x)
   (if (<= x 0.009999999776482582)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-((float) M_E) / x)), x);
	} else if (x <= 0.009999999776482582f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -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(-5.0))
		tmp = copysign(log(Float32(Float32(-Float32(exp(1))) / x)), x);
	elseif (x <= Float32(0.009999999776482582))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * 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(-5.0))
		tmp = sign(x) * abs(log((-single(2.71828182845904523536) / x)));
	elseif (x <= single(0.009999999776482582))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * 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 < -5

   1. Initial program 48.7%

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

   3. Taylor expanded in x around -inf 48.3%

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

      1. distribute-lft-out 48.3%

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

   5. Simplified 48.3%

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

      1. add-sqr-sqrt 48.3%

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

      2. sqrt-unprod 48.3%

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

      3. mul-1-neg 48.3%

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

      4. mul-1-neg 48.3%

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

      5. sqr-neg 48.3%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 48.3%

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

      8. add-log-exp 48.3%

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

      9. sum-log 48.3%

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

      10. add-sqr-sqrt -0.0%

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

      11. add-sqr-sqrt 48.3%

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

      12. add-sqr-sqrt -0.0%

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

      13. fabs-sqr -0.0%

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

      14. add-sqr-sqrt 40.4%

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

      15. *-inverses 40.4%

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

   7. Applied egg-rr 40.4%

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

      1. associate-*l/ 40.4%

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

      2. mul-1-neg 40.4%

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

   9. Simplified 40.4%

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

   if -5 < x < 0.00999999978

   1. Initial program 17.5%

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

   3. Taylor expanded in x around 0 18.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative 18.3%

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

      2. fma-define 18.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. +-commutative 18.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\left|x\right| + 1}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. rem-square-sqrt 9.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. fabs-sqr 9.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. rem-square-sqrt 18.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{x} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      7. log1p-define 99.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      8. rem-square-sqrt 52.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      9. fabs-sqr 52.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      10. rem-square-sqrt 99.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   5. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. unpow2 99.7%

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

   10. Applied egg-rr 99.7%

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

   if 0.00999999978 < x

   1. Initial program 55.9%

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

   3. Taylor expanded in x around 0 55.9%

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

      1. rem-square-sqrt 55.9%

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

      2. fabs-sqr 55.9%

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

      3. metadata-eval 55.9%

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

      4. unpow2 55.9%

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

      5. hypot-undefine 98.5%

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

      6. rem-square-sqrt 98.5%

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

   5. Simplified 98.5%

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

Alternative 5: 70.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-e}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(1 + \log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ (- E) x)) x)
   (if (<= x 0.019999999552965164)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (+ 1.0 (log x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-((float) M_E) / x)), x);
	} else if (x <= 0.019999999552965164f) {
		tmp = copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
	} else {
		tmp = copysignf((1.0f + logf(x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-Float32(exp(1))) / x)), x);
	elseif (x <= Float32(0.019999999552965164))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(Float32(Float32(1.0) + log(x)), 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((-single(2.71828182845904523536) / x)));
	elseif (x <= single(0.019999999552965164))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * single(-0.16666666666666666)))));
	else
		tmp = sign(x) * abs((single(1.0) + log(x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 48.7%

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

   3. Taylor expanded in x around -inf 48.3%

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

      1. distribute-lft-out 48.3%

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

   5. Simplified 48.3%

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

      1. add-sqr-sqrt 48.3%

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

      2. sqrt-unprod 48.3%

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

      3. mul-1-neg 48.3%

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

      4. mul-1-neg 48.3%

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

      5. sqr-neg 48.3%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 48.3%

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

      8. add-log-exp 48.3%

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

      9. sum-log 48.3%

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

      10. add-sqr-sqrt -0.0%

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

      11. add-sqr-sqrt 48.3%

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

      12. add-sqr-sqrt -0.0%

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

      13. fabs-sqr -0.0%

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

      14. add-sqr-sqrt 40.4%

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

      15. *-inverses 40.4%

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

   7. Applied egg-rr 40.4%

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

      1. associate-*l/ 40.4%

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

      2. mul-1-neg 40.4%

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

   9. Simplified 40.4%

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

   if -5 < x < 0.0199999996

   1. Initial program 18.1%

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

   3. Taylor expanded in x around 0 18.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative 18.8%

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

      2. fma-define 18.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. +-commutative 18.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\left|x\right| + 1}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. rem-square-sqrt 10.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. fabs-sqr 10.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. rem-square-sqrt 18.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{x} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      7. log1p-define 99.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      8. rem-square-sqrt 52.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      9. fabs-sqr 52.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      10. rem-square-sqrt 99.5%

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   5. Simplified 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. unpow2 99.7%

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

   10. Applied egg-rr 99.7%

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

   if 0.0199999996 < x

   1. Initial program 55.3%

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

   3. Taylor expanded in x around inf 48.3%

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

      1. +-commutative 48.3%

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

      2. rem-square-sqrt 48.3%

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

      3. fabs-sqr 48.3%

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

      4. rem-square-sqrt 48.3%

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

      5. mul-1-neg 48.3%

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

      6. log-rec 48.3%

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

      7. remove-double-neg 48.3%

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

   5. Simplified 48.3%

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

   6. Taylor expanded in x around 0 48.3%

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

Alternative 6: 62.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if x < 0.0199999996

   1. Initial program 28.2%

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

   3. Taylor expanded in x around 0 16.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative 16.0%

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

      2. fma-define 16.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. +-commutative 16.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\left|x\right| + 1}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. rem-square-sqrt 7.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. fabs-sqr 7.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. rem-square-sqrt 15.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{x} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      7. log1p-define 70.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      8. rem-square-sqrt 35.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      9. fabs-sqr 35.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      10. rem-square-sqrt 66.8%

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   5. Simplified 66.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 69.6%

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

      1. *-commutative 69.6%

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

   8. Simplified 69.6%

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

      1. unpow2 69.6%

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

   10. Applied egg-rr 69.6%

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

   if 0.0199999996 < x

   1. Initial program 55.3%

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

   3. Taylor expanded in x around inf 48.3%

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

      1. +-commutative 48.3%

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

      2. rem-square-sqrt 48.3%

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

      3. fabs-sqr 48.3%

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

      4. rem-square-sqrt 48.3%

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

      5. mul-1-neg 48.3%

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

      6. log-rec 48.3%

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

      7. remove-double-neg 48.3%

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

   5. Simplified 48.3%

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

   6. Taylor expanded in x around 0 48.3%

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

Alternative 7: 62.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.019999999552965164)
   (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
   (copysign (log (* x E)) x)))

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.019999999552965164))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(exp(1)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(0.019999999552965164))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * single(-0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x * single(2.71828182845904523536))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0199999996

   1. Initial program 28.2%

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

   3. Taylor expanded in x around 0 16.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative 16.0%

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

      2. fma-define 16.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

      3. +-commutative 16.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\left|x\right| + 1}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      4. rem-square-sqrt 7.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      5. fabs-sqr 7.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      6. rem-square-sqrt 15.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{x} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

      7. log1p-define 70.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

      8. rem-square-sqrt 35.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

      9. fabs-sqr 35.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

      10. rem-square-sqrt 66.8%

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   5. Simplified 66.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 69.6%

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

      1. *-commutative 69.6%

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

   8. Simplified 69.6%

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

      1. unpow2 69.6%

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

   10. Applied egg-rr 69.6%

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

   if 0.0199999996 < x

   1. Initial program 55.3%

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

   3. Taylor expanded in x around inf 48.3%

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

      1. +-commutative 48.3%

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

      2. rem-square-sqrt 48.3%

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

      3. fabs-sqr 48.3%

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

      4. rem-square-sqrt 48.3%

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

      5. mul-1-neg 48.3%

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

      6. log-rec 48.3%

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

      7. remove-double-neg 48.3%

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

   5. Simplified 48.3%

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

      1. add-log-exp 46.9%

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

      2. *-inverses 46.9%

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

      3. +-commutative 46.9%

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

      4. exp-sum 46.9%

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

      5. add-exp-log 46.9%

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

      6. exp-1-e 46.9%

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

   7. Applied egg-rr 46.9%

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

Alternative 8: 53.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x))

C

float code(float x) {
	return copysignf((x * (1.0f + ((x * x) * -0.16666666666666666f))), x);
}

Julia

function code(x)
	return copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-0.16666666666666666)))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * single(-0.16666666666666666)))));
end

Derivation

1. Initial program 35.3%

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

3. Taylor expanded in x around 0 14.6%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
4. Step-by-step derivation

   1. +-commutative 14.6%

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

   2. fma-define 14.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

   3. +-commutative 14.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\left|x\right| + 1}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

   4. rem-square-sqrt 7.9%

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

   5. fabs-sqr 7.9%

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

   6. rem-square-sqrt 14.5%

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{\color{blue}{x} + 1}, \log \left(1 + \left|x\right|\right)\right), x\right) \]

   7. log1p-define 54.5%

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]

   8. rem-square-sqrt 28.8%

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]

   9. fabs-sqr 28.8%

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]

   10. rem-square-sqrt 52.1%

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

5. Simplified 52.1%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

6. Taylor expanded in x around 0 53.4%

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

   1. *-commutative 53.4%

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

8. Simplified 53.4%

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

   1. unpow2 53.4%

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

10. Applied egg-rr 53.4%

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

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024179 
(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))