Rust f32::asinh

Percentage Accurate: 37.7% → 99.5%

Time: 8.0s

Alternatives: 13

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.7% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right) \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (+ x (* (fma (pow x 2.0) 0.075 -0.16666666666666666) (pow x 3.0)))
        x)
       (copysign (* 2.0 (log (sqrt (+ x (hypot 1.0 x))))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf((x + (fmaf(powf(x, 2.0f), 0.075f, -0.16666666666666666f) * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf((2.0f * logf(sqrtf((x + hypotf(1.0f, x))))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32(fma((x ^ Float32(2.0)), Float32(0.075), Float32(-0.16666666666666666)) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(Float32(Float32(2.0) * log(sqrt(Float32(x + hypot(Float32(1.0), x))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   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. Step-by-step derivation

      1. flip-+ 7.3%

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

      2. frac-2neg 7.3%

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

      3. log-div 7.3%

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

      4. add-sqr-sqrt 7.3%

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

      5. pow2 7.3%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 7.3%

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

      9. fma-define 7.3%

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

   4. Applied egg-rr 13.9%

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

      1. fma-undefine 13.9%

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

      2. unpow2 13.9%

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

      3. associate--r+ 55.5%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-sub0 100.0%

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

      9. associate--r- 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. neg-sub0 100.0%

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

   6. Simplified 100.0%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 19.7%

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

      1. expm1-log1p-u 19.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 19.1%

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

      3. *-un-lft-identity 19.1%

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

      4. *-un-lft-identity 19.1%

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

      5. +-commutative 19.1%

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

      6. hypot-1-def 19.1%

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

      7. add-sqr-sqrt 9.9%

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

      8. fabs-sqr 9.9%

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

      9. add-sqr-sqrt 19.4%

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

   4. Applied egg-rr 19.4%

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

      1. expm1-define 19.7%

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

   6. Simplified 19.7%

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

   7. Taylor expanded in x around 0 99.9%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. fma-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. unpow2 100.0%

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

      9. unpow3 100.0%

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

   9. Simplified 100.0%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 50.3%

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

      1. add-sqr-sqrt 50.3%

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

      2. pow2 50.3%

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

      3. log-pow 50.3%

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

      4. *-un-lft-identity 50.3%

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

      5. *-un-lft-identity 50.3%

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

      6. add-sqr-sqrt 50.3%

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

      7. fabs-sqr 50.3%

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

      8. add-sqr-sqrt 50.3%

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

      9. +-commutative 50.3%

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

      10. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right) + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (*
         x
         (+ (* (pow x 2.0) (- (* (pow x 2.0) 0.075) 0.16666666666666666)) 1.0))
        x)
       (copysign (* 2.0 (log (sqrt (+ x (hypot 1.0 x))))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf((x * ((powf(x, 2.0f) * ((powf(x, 2.0f) * 0.075f) - 0.16666666666666666f)) + 1.0f)), x);
	} else {
		tmp = copysignf((2.0f * logf(sqrtf((x + hypotf(1.0f, x))))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x * Float32(Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(0.075)) - Float32(0.16666666666666666))) + Float32(1.0))), x);
	else
		tmp = copysign(Float32(Float32(2.0) * log(sqrt(Float32(x + hypot(Float32(1.0), x))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.20000000298023224))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.05000000074505806))
		tmp = sign(x) * abs((x * (((x ^ single(2.0)) * (((x ^ single(2.0)) * single(0.075)) - single(0.16666666666666666))) + single(1.0))));
	else
		tmp = sign(x) * abs((single(2.0) * log(sqrt((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 58.4%

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

      1. flip-+ 7.3%

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

      2. frac-2neg 7.3%

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

      3. log-div 7.3%

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

      4. add-sqr-sqrt 7.3%

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

      5. pow2 7.3%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 7.3%

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

      9. fma-define 7.3%

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

   4. Applied egg-rr 13.9%

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

      1. fma-undefine 13.9%

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

      2. unpow2 13.9%

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

      3. associate--r+ 55.5%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-sub0 100.0%

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

      9. associate--r- 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. neg-sub0 100.0%

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

   6. Simplified 100.0%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 19.7%

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

      1. expm1-log1p-u 19.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 19.1%

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

      3. *-un-lft-identity 19.1%

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

      4. *-un-lft-identity 19.1%

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

      5. +-commutative 19.1%

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

      6. hypot-1-def 19.1%

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

      7. add-sqr-sqrt 9.9%

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

      8. fabs-sqr 9.9%

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

      9. add-sqr-sqrt 19.4%

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

   4. Applied egg-rr 19.4%

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

      1. expm1-define 19.7%

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

   6. Simplified 19.7%

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

   7. Taylor expanded in x around 0 99.9%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 50.3%

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

      1. add-sqr-sqrt 50.3%

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

      2. pow2 50.3%

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

      3. log-pow 50.3%

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

      4. *-un-lft-identity 50.3%

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

      5. *-un-lft-identity 50.3%

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

      6. add-sqr-sqrt 50.3%

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

      7. fabs-sqr 50.3%

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

      8. add-sqr-sqrt 50.3%

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

      9. +-commutative 50.3%

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

      10. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right) + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right) + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.4000000059604645)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.05000000074505806)
     (copysign
      (*
       x
       (+ (* (pow x 2.0) (- (* (pow x 2.0) 0.075) 0.16666666666666666)) 1.0))
      x)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.4000000059604645f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x * ((powf(x, 2.0f) * ((powf(x, 2.0f) * 0.075f) - 0.16666666666666666f)) + 1.0f)), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.4000000059604645))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x * Float32(Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(0.075)) - Float32(0.16666666666666666))) + Float32(1.0))), x);
	else
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.4000000059604645))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x * (((x ^ single(2.0)) * (((x ^ single(2.0)) * single(0.075)) - single(0.16666666666666666))) + single(1.0))));
	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 x < -0.400000006

   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. Step-by-step derivation

      1. flip-+ 7.3%

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

      2. frac-2neg 7.3%

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

      3. log-div 7.3%

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

      4. add-sqr-sqrt 7.3%

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

      5. pow2 7.3%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 7.3%

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

      9. fma-define 7.3%

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

   4. Applied egg-rr 13.9%

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

      1. fma-undefine 13.9%

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

      2. unpow2 13.9%

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

      3. associate--r+ 55.5%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-sub0 100.0%

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

      9. associate--r- 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. neg-sub0 100.0%

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

   6. Simplified 100.0%

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

   if -0.400000006 < x < 0.0500000007

   1. Initial program 19.7%

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

      1. expm1-log1p-u 19.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 19.1%

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

      3. *-un-lft-identity 19.1%

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

      4. *-un-lft-identity 19.1%

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

      5. +-commutative 19.1%

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

      6. hypot-1-def 19.1%

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

      7. add-sqr-sqrt 9.9%

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

      8. fabs-sqr 9.9%

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

      9. add-sqr-sqrt 19.4%

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

   4. Applied egg-rr 19.4%

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

      1. expm1-define 19.7%

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

   6. Simplified 19.7%

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

   7. Taylor expanded in x around 0 99.9%

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

   if 0.0500000007 < x

   1. Initial program 50.3%

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

      1. +-commutative 50.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right) + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right) + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.4000000059604645)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.05000000074505806)
     (copysign
      (*
       x
       (+ (* (pow x 2.0) (- (* (pow x 2.0) 0.075) 0.16666666666666666)) 1.0))
      x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.4000000059604645f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x * ((powf(x, 2.0f) * ((powf(x, 2.0f) * 0.075f) - 0.16666666666666666f)) + 1.0f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.4000000059604645))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x * Float32(Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(0.075)) - Float32(0.16666666666666666))) + Float32(1.0))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.4000000059604645))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x * (((x ^ single(2.0)) * (((x ^ single(2.0)) * single(0.075)) - single(0.16666666666666666))) + single(1.0))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.400000006

   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. Step-by-step derivation

      1. flip-+ 7.3%

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

      2. frac-2neg 7.3%

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

      3. log-div 7.3%

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

      4. add-sqr-sqrt 7.3%

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

      5. pow2 7.3%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 7.3%

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

      9. fma-define 7.3%

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

   4. Applied egg-rr 13.9%

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

      1. fma-undefine 13.9%

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

      2. unpow2 13.9%

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

      3. associate--r+ 55.5%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-sub0 100.0%

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

      9. associate--r- 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. neg-sub0 100.0%

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

   6. Simplified 100.0%

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

   if -0.400000006 < x < 0.0500000007

   1. Initial program 19.7%

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

      1. expm1-log1p-u 19.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 19.1%

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

      3. *-un-lft-identity 19.1%

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

      4. *-un-lft-identity 19.1%

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

      5. +-commutative 19.1%

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

      6. hypot-1-def 19.1%

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

      7. add-sqr-sqrt 9.9%

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

      8. fabs-sqr 9.9%

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

      9. add-sqr-sqrt 19.4%

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

   4. Applied egg-rr 19.4%

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

      1. expm1-define 19.7%

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

   6. Simplified 19.7%

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

   7. Taylor expanded in x around 0 99.9%

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

   if 0.0500000007 < x

   1. Initial program 50.3%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 50.2%

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

      3. *-un-lft-identity 50.2%

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

      4. *-un-lft-identity 50.2%

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

      5. +-commutative 50.2%

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

      6. hypot-1-def 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. fabs-sqr 99.9%

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

      9. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. expm1-define 99.9%

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

   6. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right) + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot -0.16666666666666666 + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05000000074505806)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.05000000074505806)
     (copysign (* x (+ (* (pow x 2.0) -0.16666666666666666) 1.0)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.05000000074505806f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x * ((powf(x, 2.0f) * -0.16666666666666666f) + 1.0f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05000000074505806))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x * Float32(Float32((x ^ Float32(2.0)) * Float32(-0.16666666666666666)) + Float32(1.0))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.05000000074505806))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x * (((x ^ single(2.0)) * single(-0.16666666666666666)) + single(1.0))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 58.8%

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

      1. flip-+ 8.5%

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

      2. frac-2neg 8.5%

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

      3. log-div 8.5%

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

      4. add-sqr-sqrt 8.4%

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

      5. pow2 8.4%

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

      6. add-sqr-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. add-sqr-sqrt 8.4%

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

      9. fma-define 8.4%

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

   4. Applied egg-rr 14.9%

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

      1. fma-undefine 14.9%

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

      2. unpow2 14.9%

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

      3. associate--r+ 56.0%

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

      4. +-inverses 99.8%

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

      5. metadata-eval 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. neg-sub0 99.8%

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

      9. associate--r- 99.8%

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

      10. neg-sub0 99.8%

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

      11. +-commutative 99.8%

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

      12. sub-neg 99.8%

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

      13. neg-sub0 99.8%

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

   6. Simplified 99.8%

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

   if -0.0500000007 < x < 0.0500000007

   1. Initial program 19.1%

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

      1. expm1-log1p-u 19.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 18.5%

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

      3. *-un-lft-identity 18.5%

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

      4. *-un-lft-identity 18.5%

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

      5. +-commutative 18.5%

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

      6. hypot-1-def 18.5%

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

      7. add-sqr-sqrt 9.9%

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

      8. fabs-sqr 9.9%

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

      9. add-sqr-sqrt 18.8%

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

   4. Applied egg-rr 18.8%

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

      1. expm1-define 19.1%

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

   6. Simplified 19.1%

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

   7. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   if 0.0500000007 < x

   1. Initial program 50.3%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 50.2%

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

      3. *-un-lft-identity 50.2%

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

      4. *-un-lft-identity 50.2%

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

      5. +-commutative 50.2%

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

      6. hypot-1-def 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. fabs-sqr 99.9%

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

      9. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. expm1-define 99.9%

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

   6. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot -0.16666666666666666 + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot -0.16666666666666666 + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (* x (+ (* (pow x 2.0) -0.16666666666666666) 1.0)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x * ((powf(x, 2.0f) * -0.16666666666666666f) + 1.0f)), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x * Float32(Float32((x ^ Float32(2.0)) * Float32(-0.16666666666666666)) + Float32(1.0))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x * (((x ^ single(2.0)) * single(-0.16666666666666666)) + single(1.0))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. mul-1-neg 99.2%

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

      4. unsub-neg 99.2%

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

      5. sub-neg 99.2%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 11.4%

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

      9. associate-*r/ 11.4%

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

      10. metadata-eval 11.4%

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

      11. distribute-neg-frac 11.4%

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

      12. metadata-eval 11.4%

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

   5. Simplified 11.4%

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

   6. Taylor expanded in x around 0 98.0%

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

   if -2 < x < 0.0500000007

   1. Initial program 20.3%

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

      1. expm1-log1p-u 20.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 19.7%

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

      3. *-un-lft-identity 19.7%

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

      4. *-un-lft-identity 19.7%

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

      5. +-commutative 19.7%

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

      6. hypot-1-def 19.7%

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

      7. add-sqr-sqrt 9.8%

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

      8. fabs-sqr 9.8%

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

      9. add-sqr-sqrt 20.0%

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

   4. Applied egg-rr 20.0%

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

      1. expm1-define 20.4%

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

   6. Simplified 20.4%

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

   7. Taylor expanded in x around 0 99.4%

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

      1. *-commutative 99.4%

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

   9. Simplified 99.4%

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

   if 0.0500000007 < x

   1. Initial program 50.3%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 50.2%

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

      3. *-un-lft-identity 50.2%

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

      4. *-un-lft-identity 50.2%

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

      5. +-commutative 50.2%

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

      6. hypot-1-def 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. fabs-sqr 99.9%

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

      9. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. expm1-define 99.9%

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

   6. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot -0.16666666666666666 + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot -0.16666666666666666 + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (* x (+ (* (pow x 2.0) -0.16666666666666666) 1.0)) x)
     (copysign (log (+ x (fabs x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x * ((powf(x, 2.0f) * -0.16666666666666666f) + 1.0f)), x);
	} else {
		tmp = copysignf(logf((x + fabsf(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x * Float32(Float32((x ^ Float32(2.0)) * Float32(-0.16666666666666666)) + Float32(1.0))), x);
	else
		tmp = copysign(log(Float32(x + abs(x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x * (((x ^ single(2.0)) * single(-0.16666666666666666)) + single(1.0))));
	else
		tmp = sign(x) * abs(log((x + abs(x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. mul-1-neg 99.2%

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

      4. unsub-neg 99.2%

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

      5. sub-neg 99.2%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 11.4%

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

      9. associate-*r/ 11.4%

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

      10. metadata-eval 11.4%

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

      11. distribute-neg-frac 11.4%

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

      12. metadata-eval 11.4%

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

   5. Simplified 11.4%

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

   6. Taylor expanded in x around 0 98.0%

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

   if -2 < x < 0.0500000007

   1. Initial program 20.3%

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

      1. expm1-log1p-u 20.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 19.7%

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

      3. *-un-lft-identity 19.7%

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

      4. *-un-lft-identity 19.7%

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

      5. +-commutative 19.7%

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

      6. hypot-1-def 19.7%

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

      7. add-sqr-sqrt 9.8%

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

      8. fabs-sqr 9.8%

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

      9. add-sqr-sqrt 20.0%

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

   4. Applied egg-rr 20.0%

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

      1. expm1-define 20.4%

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

   6. Simplified 20.4%

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

   7. Taylor expanded in x around 0 99.4%

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

      1. *-commutative 99.4%

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

   9. Simplified 99.4%

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

   if 0.0500000007 < x

   1. Initial program 50.3%

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

   3. Taylor expanded in x around inf 97.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot -0.16666666666666666 + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot -0.16666666666666666 + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (* x (+ (* (pow x 2.0) -0.16666666666666666) 1.0)) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x * ((powf(x, 2.0f) * -0.16666666666666666f) + 1.0f)), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x * Float32(Float32((x ^ Float32(2.0)) * Float32(-0.16666666666666666)) + Float32(1.0))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x * (((x ^ single(2.0)) * single(-0.16666666666666666)) + single(1.0))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. mul-1-neg 99.2%

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

      4. unsub-neg 99.2%

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

      5. sub-neg 99.2%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 11.4%

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

      9. associate-*r/ 11.4%

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

      10. metadata-eval 11.4%

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

      11. distribute-neg-frac 11.4%

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

      12. metadata-eval 11.4%

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

   5. Simplified 11.4%

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

   6. Taylor expanded in x around 0 98.0%

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

   if -2 < x < 0.0500000007

   1. Initial program 20.3%

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

      1. expm1-log1p-u 20.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 19.7%

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

      3. *-un-lft-identity 19.7%

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

      4. *-un-lft-identity 19.7%

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

      5. +-commutative 19.7%

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

      6. hypot-1-def 19.7%

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

      7. add-sqr-sqrt 9.8%

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

      8. fabs-sqr 9.8%

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

      9. add-sqr-sqrt 20.0%

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

   4. Applied egg-rr 20.0%

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

      1. expm1-define 20.4%

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

   6. Simplified 20.4%

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

   7. Taylor expanded in x around 0 99.4%

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

      1. *-commutative 99.4%

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

   9. Simplified 99.4%

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

   if 0.0500000007 < x

   1. Initial program 50.3%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 50.2%

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

      3. *-un-lft-identity 50.2%

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

      4. *-un-lft-identity 50.2%

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

      5. +-commutative 50.2%

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

      6. hypot-1-def 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. fabs-sqr 99.9%

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

      9. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. expm1-define 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 97.3%

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

      1. *-commutative 97.3%

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

   9. 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. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot -0.16666666666666666 + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. mul-1-neg 99.2%

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

      4. unsub-neg 99.2%

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

      5. sub-neg 99.2%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 11.4%

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

      9. associate-*r/ 11.4%

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

      10. metadata-eval 11.4%

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

      11. distribute-neg-frac 11.4%

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

      12. metadata-eval 11.4%

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

   5. Simplified 11.4%

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

   6. Taylor expanded in x around 0 98.0%

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

   if -2 < x < 0.0500000007

   1. Initial program 20.3%

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

      1. expm1-log1p-u 20.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 19.7%

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

      3. *-un-lft-identity 19.7%

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

      4. *-un-lft-identity 19.7%

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

      5. +-commutative 19.7%

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

      6. hypot-1-def 19.7%

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

      7. add-sqr-sqrt 9.8%

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

      8. fabs-sqr 9.8%

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

      9. add-sqr-sqrt 20.0%

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

   4. Applied egg-rr 20.0%

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

      1. expm1-define 20.4%

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

   6. Simplified 20.4%

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

   7. Taylor expanded in x around 0 99.4%

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

      1. distribute-rgt-in 99.4%

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

      2. *-lft-identity 99.4%

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

      3. associate-*l* 99.4%

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

      4. unpow2 99.4%

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

      5. unpow3 99.4%

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

   9. Simplified 99.4%

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

   if 0.0500000007 < x

   1. Initial program 50.3%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 50.2%

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

      3. *-un-lft-identity 50.2%

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

      4. *-un-lft-identity 50.2%

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

      5. +-commutative 50.2%

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

      6. hypot-1-def 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. fabs-sqr 99.9%

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

      9. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. expm1-define 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 97.3%

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

      1. *-commutative 97.3%

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

   9. 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: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign x x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. mul-1-neg 99.2%

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

      4. unsub-neg 99.2%

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

      5. sub-neg 99.2%

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

      6. rem-square-sqrt -0.0%

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

      7. fabs-sqr -0.0%

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

      8. rem-square-sqrt 11.4%

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

      9. associate-*r/ 11.4%

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

      10. metadata-eval 11.4%

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

      11. distribute-neg-frac 11.4%

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

      12. metadata-eval 11.4%

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

   5. Simplified 11.4%

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

   6. Taylor expanded in x around 0 98.0%

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

   if -2 < x < 0.0500000007

   1. Initial program 20.3%

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

      1. flip-+ 20.3%

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

      2. frac-2neg 20.3%

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

      3. log-div 20.6%

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

      4. add-sqr-sqrt 20.6%

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

      5. pow2 20.6%

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

      6. add-sqr-sqrt 10.3%

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

      7. fabs-sqr 10.3%

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

      8. add-sqr-sqrt 20.6%

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

      9. fma-define 20.6%

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

   4. Applied egg-rr 20.6%

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

      1. fma-undefine 20.6%

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

      2. unpow2 20.6%

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

      3. associate--r+ 20.6%

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

      4. +-inverses 20.6%

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

      5. metadata-eval 20.6%

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

      6. metadata-eval 20.6%

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

      7. metadata-eval 20.6%

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

      8. neg-sub0 20.6%

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

      9. associate--r- 20.6%

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

      10. neg-sub0 20.6%

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

      11. +-commutative 20.6%

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

      12. sub-neg 20.6%

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

      13. neg-sub0 20.6%

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

   6. Simplified 20.6%

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

   7. Taylor expanded in x around 0 98.1%

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

      1. neg-mul-1 98.1%

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

   9. Simplified 98.1%

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

   if 0.0500000007 < x

   1. Initial program 50.3%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 50.2%

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

      3. *-un-lft-identity 50.2%

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

      4. *-un-lft-identity 50.2%

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

      5. +-commutative 50.2%

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

      6. hypot-1-def 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. fabs-sqr 99.9%

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

      9. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. expm1-define 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 97.3%

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

      1. *-commutative 97.3%

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

   9. 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. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.05000000074505806) (copysign x x) (copysign (log (* x 2.0)) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.05000000074505806f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.05000000074505806))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(0.05000000074505806))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0500000007

   1. Initial program 34.2%

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

      1. expm1-log1p-u 34.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 33.7%

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

      3. *-un-lft-identity 33.7%

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

      4. *-un-lft-identity 33.7%

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

      5. +-commutative 33.7%

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

      6. hypot-1-def 49.3%

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

      7. add-sqr-sqrt 6.2%

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

      8. fabs-sqr 6.2%

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

      9. add-sqr-sqrt 17.2%

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

   4. Applied egg-rr 17.2%

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

      1. expm1-define 17.4%

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

   6. Simplified 17.4%

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

   7. Taylor expanded in x around 0 65.7%

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

      1. distribute-rgt-in 65.7%

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

      2. *-lft-identity 65.7%

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

      3. associate-*l* 65.7%

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

      4. unpow2 65.7%

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

      5. unpow3 65.7%

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

   9. Simplified 65.7%

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

   10. Taylor expanded in x around 0 66.1%

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

   if 0.0500000007 < x

   1. Initial program 50.3%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 50.2%

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

      3. *-un-lft-identity 50.2%

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

      4. *-un-lft-identity 50.2%

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

      5. +-commutative 50.2%

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

      6. hypot-1-def 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. fabs-sqr 99.9%

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

      9. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. expm1-define 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 97.3%

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

      1. *-commutative 97.3%

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

   9. Simplified 97.3%

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

Alternative 12: 62.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.05000000074505806) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.05000000074505806f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.05000000074505806))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0500000007

   1. Initial program 34.2%

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

      1. expm1-log1p-u 34.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-undefine 33.7%

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

      3. *-un-lft-identity 33.7%

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

      4. *-un-lft-identity 33.7%

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

      5. +-commutative 33.7%

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

      6. hypot-1-def 49.3%

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

      7. add-sqr-sqrt 6.2%

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

      8. fabs-sqr 6.2%

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

      9. add-sqr-sqrt 17.2%

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

   4. Applied egg-rr 17.2%

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

      1. expm1-define 17.4%

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

   6. Simplified 17.4%

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

   7. Taylor expanded in x around 0 65.7%

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

      1. distribute-rgt-in 65.7%

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

      2. *-lft-identity 65.7%

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

      3. associate-*l* 65.7%

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

      4. unpow2 65.7%

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

      5. unpow3 65.7%

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

   9. Simplified 65.7%

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

   10. Taylor expanded in x around 0 66.1%

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

   if 0.0500000007 < x

   1. Initial program 50.3%

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

   3. Taylor expanded in x around 0 44.2%

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

      1. log1p-define 44.2%

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

      2. rem-square-sqrt 44.2%

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

      3. fabs-sqr 44.2%

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

      4. rem-square-sqrt 44.2%

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

   5. Simplified 44.2%

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

Alternative 13: 54.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 38.2%

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

   1. expm1-log1p-u 38.1%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

   2. expm1-undefine 37.9%

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

   3. *-un-lft-identity 37.9%

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

   4. *-un-lft-identity 37.9%

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

   5. +-commutative 37.9%

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

   6. hypot-1-def 62.0%

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

   7. add-sqr-sqrt 29.6%

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

   8. fabs-sqr 29.6%

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

   9. add-sqr-sqrt 37.9%

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

4. Applied egg-rr 37.9%

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

   1. expm1-define 38.0%

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

6. Simplified 38.0%

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

7. Taylor expanded in x around 0 51.3%

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

   1. distribute-rgt-in 51.3%

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

   2. *-lft-identity 51.3%

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

   3. associate-*l* 51.3%

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

   4. unpow2 51.3%

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

   5. unpow3 51.3%

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

9. Simplified 51.3%

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

10. Taylor expanded in x around 0 52.4%

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

Developer target: 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 2024103 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))