Rust f32::asinh

Percentage Accurate: 37.6% → 99.5%

Time: 7.5s

Alternatives: 14

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: 37.6% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \log \left(\mathsf{hypot}\left(\sqrt{\mathsf{hypot}\left(1, x\right)}, \sqrt{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 0.10000000149011612)
       (copysign
        (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
        x)
       (copysign (* 2.0 (log (hypot (sqrt (hypot 1.0 x)) (sqrt 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 <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + (x + (0.075f * powf(x, 5.0f)))), x);
	} else {
		tmp = copysignf((2.0f * logf(hypotf(sqrtf(hypotf(1.0f, x)), sqrtf(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(-1.0))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(x + Float32(Float32(0.075) * (x ^ Float32(5.0))))), x);
	else
		tmp = copysign(Float32(Float32(2.0) * log(hypot(sqrt(hypot(Float32(1.0), x)), sqrt(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(-1.0))
		tmp = sign(x) * abs(log((abs(x) + hypot(single(1.0), x))));
	elseif (t_0 <= single(0.10000000149011612))
		tmp = sign(x) * abs(((single(-0.16666666666666666) * (x ^ single(3.0))) + (x + (single(0.075) * (x ^ single(5.0))))));
	else
		tmp = sign(x) * abs((single(2.0) * log(hypot(sqrt(hypot(single(1.0), x)), sqrt(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) 1)))) x) < -1

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

      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%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.100000001

   1. Initial program 17.4%

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

      1. log1p-expm1-u 17.4%

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

      2. expm1-udef 17.4%

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

      3. add-exp-log 17.4%

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

      4. add-sqr-sqrt 9.9%

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

      5. fabs-sqr 9.9%

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

      6. add-sqr-sqrt 17.5%

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

      7. +-commutative 17.5%

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

      8. hypot-1-def 17.4%

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

   3. Applied egg-rr 17.4%

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

      1. sub-neg 17.4%

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

      2. +-commutative 17.4%

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

      3. metadata-eval 17.4%

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

      4. associate-+l+ 17.5%

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

   5. Applied egg-rr 17.5%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)}, x\right) \]

   if 0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 50.5%

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

      1. add-sqr-sqrt 50.4%

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

      2. pow2 50.4%

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

      3. log-pow 50.4%

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

      4. add-sqr-sqrt 50.4%

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

      5. fabs-sqr 50.4%

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

      6. add-sqr-sqrt 50.4%

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

      7. +-commutative 50.4%

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

      8. hypot-1-def 95.8%

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

   3. Applied egg-rr 95.8%

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

      1. +-commutative 95.8%

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

      2. add-sqr-sqrt 95.7%

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

      3. add-sqr-sqrt 95.8%

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

      4. hypot-def 99.7%

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

   5. Applied egg-rr 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \log \left(\mathsf{hypot}\left(\sqrt{\mathsf{hypot}\left(1, x\right)}, \sqrt{x}\right)\right), x\right)\\ \end{array} \]

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.800000011920929:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log 2 + \left(\log x + \frac{0.25}{x \cdot x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 0.800000011920929)
       (copysign
        (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
        x)
       (copysign (+ (log 2.0) (+ (log x) (/ 0.25 (* x 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 <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else if (t_0 <= 0.800000011920929f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + (x + (0.075f * powf(x, 5.0f)))), x);
	} else {
		tmp = copysignf((logf(2.0f) + (logf(x) + (0.25f / (x * 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(-1.0))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	elseif (t_0 <= Float32(0.800000011920929))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(x + Float32(Float32(0.075) * (x ^ Float32(5.0))))), x);
	else
		tmp = copysign(Float32(log(Float32(2.0)) + Float32(log(x) + Float32(Float32(0.25) / Float32(x * 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(-1.0))
		tmp = sign(x) * abs(log((abs(x) + hypot(single(1.0), x))));
	elseif (t_0 <= single(0.800000011920929))
		tmp = sign(x) * abs(((single(-0.16666666666666666) * (x ^ single(3.0))) + (x + (single(0.075) * (x ^ single(5.0))))));
	else
		tmp = sign(x) * abs((log(single(2.0)) + (log(x) + (single(0.25) / (x * 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) 1)))) x) < -1

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

      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%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.800000012

   1. Initial program 18.7%

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

      1. log1p-expm1-u 18.7%

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

      2. expm1-udef 18.7%

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

      3. add-exp-log 18.7%

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

      4. add-sqr-sqrt 11.3%

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

      5. fabs-sqr 11.3%

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

      6. add-sqr-sqrt 18.8%

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

      7. +-commutative 18.8%

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

      8. hypot-1-def 18.7%

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

   3. Applied egg-rr 18.7%

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

      1. sub-neg 18.7%

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

      2. +-commutative 18.7%

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

      3. metadata-eval 18.7%

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

      4. associate-+l+ 18.8%

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

   5. Applied egg-rr 18.8%

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

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)}, x\right) \]

   if 0.800000012 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 49.2%

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

      1. *-un-lft-identity 49.2%

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

      2. *-commutative 49.2%

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

      3. log-prod 49.2%

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

      4. add-sqr-sqrt 49.1%

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

      5. fabs-sqr 49.1%

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

      6. add-sqr-sqrt 49.2%

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

      7. +-commutative 49.2%

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

      8. hypot-1-def 95.9%

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

      9. metadata-eval 95.9%

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

   3. Applied egg-rr 95.9%

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

   4. Taylor expanded in x around inf 97.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} + 0, x\right) \]
   5. Step-by-step derivation

      1. associate-+r+ 97.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)} + 0, x\right) \]

      2. +-commutative 97.5%

         \[\leadsto \mathsf{copysign}\left(\left(\color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right)} + 0.25 \cdot \frac{1}{{x}^{2}}\right) + 0, x\right) \]

      3. *-lft-identity 97.5%

         \[\leadsto \mathsf{copysign}\left(\left(\color{blue}{1 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right)} + 0.25 \cdot \frac{1}{{x}^{2}}\right) + 0, x\right) \]

      4. distribute-lft-in 97.5%

         \[\leadsto \mathsf{copysign}\left(\left(\color{blue}{\left(1 \cdot \log 2 + 1 \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} + 0.25 \cdot \frac{1}{{x}^{2}}\right) + 0, x\right) \]

      5. *-lft-identity 97.5%

         \[\leadsto \mathsf{copysign}\left(\left(\left(\color{blue}{\log 2} + 1 \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right)\right)\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right) + 0, x\right) \]

      6. associate-+l+ 97.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 2 + \left(1 \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right)\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} + 0, x\right) \]

      7. log-rec 97.4%

         \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(1 \cdot \left(-1 \cdot \color{blue}{\left(-\log x\right)}\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right) + 0, x\right) \]

      8. mul-1-neg 97.4%

         \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(1 \cdot \color{blue}{\left(-\left(-\log x\right)\right)} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right) + 0, x\right) \]

      9. remove-double-neg 97.4%

         \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(1 \cdot \color{blue}{\log x} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right) + 0, x\right) \]

      10. *-lft-identity 97.4%

          \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(\color{blue}{\log x} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right) + 0, x\right) \]

      11. associate-*r/ 97.4%

          \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(\log x + \color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) + 0, x\right) \]

      12. metadata-eval 97.4%

          \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(\log x + \frac{\color{blue}{0.25}}{{x}^{2}}\right)\right) + 0, x\right) \]

      13. unpow2 97.4%

          \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(\log x + \frac{0.25}{\color{blue}{x \cdot x}}\right)\right) + 0, x\right) \]

   6. Simplified 97.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 2 + \left(\log x + \frac{0.25}{x \cdot x}\right)\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.800000011920929:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log 2 + \left(\log x + \frac{0.25}{x \cdot x}\right), x\right)\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) - 0.5 \cdot \frac{1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- (- (fabs x) x) (* 0.5 (/ 1.0 x)))) x)
   (if (<= x 0.10000000149011612)
     (copysign
      (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
      x)
     (copysign (log1p (+ (hypot 1.0 x) (+ x -1.0))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) - (0.5f * (1.0f / x)))), x);
	} else if (x <= 0.10000000149011612f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + (x + (0.075f * powf(x, 5.0f)))), x);
	} else {
		tmp = copysignf(log1pf((hypotf(1.0f, x) + (x + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) - Float32(Float32(0.5) * Float32(Float32(1.0) / x)))), x);
	elseif (x <= Float32(0.10000000149011612))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(x + Float32(Float32(0.075) * (x ^ Float32(5.0))))), x);
	else
		tmp = copysign(log1p(Float32(hypot(Float32(1.0), x) + Float32(x + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 59.9%

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

   2. Taylor expanded in x around -inf 99.3%

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

   if -2 < x < 0.100000001

   1. Initial program 17.4%

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

      1. log1p-expm1-u 17.4%

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

      2. expm1-udef 17.4%

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

      3. add-exp-log 17.4%

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

      4. add-sqr-sqrt 9.9%

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

      5. fabs-sqr 9.9%

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

      6. add-sqr-sqrt 17.5%

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

      7. +-commutative 17.5%

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

      8. hypot-1-def 17.4%

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

   3. Applied egg-rr 17.4%

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

      1. sub-neg 17.4%

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

      2. +-commutative 17.4%

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

      3. metadata-eval 17.4%

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

      4. associate-+l+ 17.5%

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

   5. Applied egg-rr 17.5%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)}, x\right) \]

   if 0.100000001 < x

   1. Initial program 50.5%

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

      1. log1p-expm1-u 50.5%

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

      2. expm1-udef 50.5%

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

      3. add-exp-log 50.5%

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

      4. add-sqr-sqrt 50.5%

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

      5. fabs-sqr 50.5%

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

      6. add-sqr-sqrt 50.5%

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

      7. +-commutative 50.5%

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

      8. hypot-1-def 96.0%

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

   3. Applied egg-rr 96.0%

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

      1. sub-neg 96.0%

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

      2. +-commutative 96.0%

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

      3. metadata-eval 96.0%

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

      4. associate-+l+ 96.0%

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

   5. Applied egg-rr 96.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) - 0.5 \cdot \frac{1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) - 0.5 \cdot \frac{1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log 2 + \left(\log x + \frac{0.25}{x \cdot x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- (- (fabs x) x) (* 0.5 (/ 1.0 x)))) x)
   (if (<= x 1.0)
     (copysign
      (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
      x)
     (copysign (+ (log 2.0) (+ (log x) (/ 0.25 (* x x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) - (0.5f * (1.0f / x)))), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + (x + (0.075f * powf(x, 5.0f)))), x);
	} else {
		tmp = copysignf((logf(2.0f) + (logf(x) + (0.25f / (x * x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) - Float32(Float32(0.5) * Float32(Float32(1.0) / x)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(x + Float32(Float32(0.075) * (x ^ Float32(5.0))))), x);
	else
		tmp = copysign(Float32(log(Float32(2.0)) + Float32(log(x) + Float32(Float32(0.25) / Float32(x * 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(((abs(x) - x) - (single(0.5) * (single(1.0) / x)))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(((single(-0.16666666666666666) * (x ^ single(3.0))) + (x + (single(0.075) * (x ^ single(5.0))))));
	else
		tmp = sign(x) * abs((log(single(2.0)) + (log(x) + (single(0.25) / (x * x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 59.9%

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

   2. Taylor expanded in x around -inf 99.3%

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

   if -2 < x < 1

   1. Initial program 18.7%

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

      1. log1p-expm1-u 18.7%

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

      2. expm1-udef 18.7%

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

      3. add-exp-log 18.7%

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

      4. add-sqr-sqrt 11.3%

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

      5. fabs-sqr 11.3%

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

      6. add-sqr-sqrt 18.8%

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

      7. +-commutative 18.8%

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

      8. hypot-1-def 18.7%

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

   3. Applied egg-rr 18.7%

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

      1. sub-neg 18.7%

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

      2. +-commutative 18.7%

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

      3. metadata-eval 18.7%

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

      4. associate-+l+ 18.8%

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

   5. Applied egg-rr 18.8%

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

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)}, x\right) \]

   if 1 < x

   1. Initial program 49.2%

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

      1. *-un-lft-identity 49.2%

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

      2. *-commutative 49.2%

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

      3. log-prod 49.2%

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

      4. add-sqr-sqrt 49.1%

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

      5. fabs-sqr 49.1%

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

      6. add-sqr-sqrt 49.2%

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

      7. +-commutative 49.2%

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

      8. hypot-1-def 95.9%

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

      9. metadata-eval 95.9%

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

   3. Applied egg-rr 95.9%

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

   4. Taylor expanded in x around inf 97.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} + 0, x\right) \]
   5. Step-by-step derivation

      1. associate-+r+ 97.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)} + 0, x\right) \]

      2. +-commutative 97.5%

         \[\leadsto \mathsf{copysign}\left(\left(\color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right)} + 0.25 \cdot \frac{1}{{x}^{2}}\right) + 0, x\right) \]

      3. *-lft-identity 97.5%

         \[\leadsto \mathsf{copysign}\left(\left(\color{blue}{1 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right)} + 0.25 \cdot \frac{1}{{x}^{2}}\right) + 0, x\right) \]

      4. distribute-lft-in 97.5%

         \[\leadsto \mathsf{copysign}\left(\left(\color{blue}{\left(1 \cdot \log 2 + 1 \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} + 0.25 \cdot \frac{1}{{x}^{2}}\right) + 0, x\right) \]

      5. *-lft-identity 97.5%

         \[\leadsto \mathsf{copysign}\left(\left(\left(\color{blue}{\log 2} + 1 \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right)\right)\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right) + 0, x\right) \]

      6. associate-+l+ 97.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 2 + \left(1 \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right)\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} + 0, x\right) \]

      7. log-rec 97.4%

         \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(1 \cdot \left(-1 \cdot \color{blue}{\left(-\log x\right)}\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right) + 0, x\right) \]

      8. mul-1-neg 97.4%

         \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(1 \cdot \color{blue}{\left(-\left(-\log x\right)\right)} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right) + 0, x\right) \]

      9. remove-double-neg 97.4%

         \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(1 \cdot \color{blue}{\log x} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right) + 0, x\right) \]

      10. *-lft-identity 97.4%

          \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(\color{blue}{\log x} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right) + 0, x\right) \]

      11. associate-*r/ 97.4%

          \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(\log x + \color{blue}{\frac{0.25 \cdot 1}{{x}^{2}}}\right)\right) + 0, x\right) \]

      12. metadata-eval 97.4%

          \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(\log x + \frac{\color{blue}{0.25}}{{x}^{2}}\right)\right) + 0, x\right) \]

      13. unpow2 97.4%

          \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \left(\log x + \frac{0.25}{\color{blue}{x \cdot x}}\right)\right) + 0, x\right) \]

   6. Simplified 97.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 2 + \left(\log x + \frac{0.25}{x \cdot x}\right)\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) - 0.5 \cdot \frac{1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log 2 + \left(\log x + \frac{0.25}{x \cdot x}\right), x\right)\\ \end{array} \]

Alternative 5: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) - 0.5 \cdot \frac{1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- (- (fabs x) x) (* 0.5 (/ 1.0 x)))) x)
   (if (<= x 0.004999999888241291)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log1p (+ (hypot 1.0 x) (+ x -1.0))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) - (0.5f * (1.0f / x)))), x);
	} else if (x <= 0.004999999888241291f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(log1pf((hypotf(1.0f, x) + (x + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) - Float32(Float32(0.5) * Float32(Float32(1.0) / x)))), x);
	elseif (x <= Float32(0.004999999888241291))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log1p(Float32(hypot(Float32(1.0), x) + Float32(x + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 59.9%

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

   2. Taylor expanded in x around -inf 99.3%

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

   if -2 < x < 0.00499999989

   1. Initial program 16.8%

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

      1. log1p-expm1-u 16.8%

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

      2. expm1-udef 16.8%

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

      3. add-exp-log 16.8%

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

      4. add-sqr-sqrt 9.2%

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

      5. fabs-sqr 9.2%

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

      6. add-sqr-sqrt 16.8%

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

      7. +-commutative 16.8%

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

      8. hypot-1-def 16.8%

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

   3. Applied egg-rr 16.8%

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

      1. sub-neg 16.8%

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

      2. +-commutative 16.8%

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

      3. metadata-eval 16.8%

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

      4. associate-+l+ 16.9%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in x around 0 100.0%

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

   if 0.00499999989 < x

   1. Initial program 51.1%

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

      1. log1p-expm1-u 51.1%

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

      2. expm1-udef 51.1%

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

      3. add-exp-log 51.1%

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

      4. add-sqr-sqrt 51.1%

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

      5. fabs-sqr 51.1%

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

      6. add-sqr-sqrt 51.1%

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

      7. +-commutative 51.1%

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

      8. hypot-1-def 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. sub-neg 95.9%

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

      2. +-commutative 95.9%

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

      3. metadata-eval 95.9%

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

      4. associate-+l+ 95.9%

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

   5. Applied egg-rr 95.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) - 0.5 \cdot \frac{1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \]

Alternative 6: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.004999999888241291)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log1p (+ (hypot 1.0 x) (+ x -1.0))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.004999999888241291f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(log1pf((hypotf(1.0f, x) + (x + -1.0f))), 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(0.004999999888241291))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log1p(Float32(hypot(Float32(1.0), x) + Float32(x + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. *-commutative 59.9%

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

      3. log-prod 59.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 12.6%

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

      7. +-commutative 12.6%

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

      8. hypot-1-def 12.6%

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

      9. metadata-eval 12.6%

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

   3. Applied egg-rr 12.6%

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

   4. Taylor expanded in x around -inf 97.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)} + 0, x\right) \]

   if -2 < x < 0.00499999989

   1. Initial program 16.8%

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

      1. log1p-expm1-u 16.8%

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

      2. expm1-udef 16.8%

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

      3. add-exp-log 16.8%

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

      4. add-sqr-sqrt 9.2%

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

      5. fabs-sqr 9.2%

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

      6. add-sqr-sqrt 16.8%

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

      7. +-commutative 16.8%

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

      8. hypot-1-def 16.8%

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

   3. Applied egg-rr 16.8%

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

      1. sub-neg 16.8%

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

      2. +-commutative 16.8%

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

      3. metadata-eval 16.8%

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

      4. associate-+l+ 16.9%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in x around 0 100.0%

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

   if 0.00499999989 < x

   1. Initial program 51.1%

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

      1. log1p-expm1-u 51.1%

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

      2. expm1-udef 51.1%

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

      3. add-exp-log 51.1%

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

      4. add-sqr-sqrt 51.1%

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

      5. fabs-sqr 51.1%

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

      6. add-sqr-sqrt 51.1%

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

      7. +-commutative 51.1%

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

      8. hypot-1-def 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. sub-neg 95.9%

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

      2. +-commutative 95.9%

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

      3. metadata-eval 95.9%

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

      4. associate-+l+ 95.9%

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

   5. Applied egg-rr 95.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \]

Alternative 7: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- (/ 0.125 (pow x 3.0)) (/ 0.5 x))) x)
   (if (<= x 0.004999999888241291)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log1p (+ (hypot 1.0 x) (+ x -1.0))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((0.125f / powf(x, 3.0f)) - (0.5f / x))), x);
	} else if (x <= 0.004999999888241291f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(log1pf((hypotf(1.0f, x) + (x + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(Float32(0.125) / (x ^ Float32(3.0))) - Float32(Float32(0.5) / x))), x);
	elseif (x <= Float32(0.004999999888241291))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log1p(Float32(hypot(Float32(1.0), x) + Float32(x + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. *-commutative 59.9%

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

      3. log-prod 59.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 12.6%

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

      7. +-commutative 12.6%

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

      8. hypot-1-def 12.6%

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

      9. metadata-eval 12.6%

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

   3. Applied egg-rr 12.6%

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

   4. Taylor expanded in x around -inf 99.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)} + 0, x\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 99.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}} - 0.5 \cdot \frac{1}{x}\right) + 0, x\right) \]

      2. metadata-eval 99.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0.125}}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right) + 0, x\right) \]

      3. associate-*r/ 99.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{0.125}{{x}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) + 0, x\right) \]

      4. metadata-eval 99.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{0.125}{{x}^{3}} - \frac{\color{blue}{0.5}}{x}\right) + 0, x\right) \]

   6. Simplified 99.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right)} + 0, x\right) \]

   if -2 < x < 0.00499999989

   1. Initial program 16.8%

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

      1. log1p-expm1-u 16.8%

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

      2. expm1-udef 16.8%

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

      3. add-exp-log 16.8%

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

      4. add-sqr-sqrt 9.2%

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

      5. fabs-sqr 9.2%

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

      6. add-sqr-sqrt 16.8%

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

      7. +-commutative 16.8%

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

      8. hypot-1-def 16.8%

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

   3. Applied egg-rr 16.8%

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

      1. sub-neg 16.8%

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

      2. +-commutative 16.8%

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

      3. metadata-eval 16.8%

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

      4. associate-+l+ 16.9%

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

   5. Applied egg-rr 16.9%

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

   6. Taylor expanded in x around 0 100.0%

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

   if 0.00499999989 < x

   1. Initial program 51.1%

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

      1. log1p-expm1-u 51.1%

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

      2. expm1-udef 51.1%

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

      3. add-exp-log 51.1%

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

      4. add-sqr-sqrt 51.1%

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

      5. fabs-sqr 51.1%

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

      6. add-sqr-sqrt 51.1%

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

      7. +-commutative 51.1%

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

      8. hypot-1-def 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. sub-neg 95.9%

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

      2. +-commutative 95.9%

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

      3. metadata-eval 95.9%

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

      4. associate-+l+ 95.9%

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

   5. Applied egg-rr 95.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \]

Alternative 8: 98.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 2.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf((logf(2.0f) + logf(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(2.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(Float32(log(Float32(2.0)) + log(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(2.0))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs((log(single(2.0)) + log(x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. *-commutative 59.9%

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

      3. log-prod 59.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 12.6%

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

      7. +-commutative 12.6%

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

      8. hypot-1-def 12.6%

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

      9. metadata-eval 12.6%

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

   3. Applied egg-rr 12.6%

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

   4. Taylor expanded in x around -inf 97.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)} + 0, x\right) \]

   if -2 < x < 2

   1. Initial program 19.4%

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

      1. log1p-expm1-u 19.4%

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

      2. expm1-udef 19.4%

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

      3. add-exp-log 19.4%

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

      4. add-sqr-sqrt 12.0%

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

      5. fabs-sqr 12.0%

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

      6. add-sqr-sqrt 19.4%

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

      7. +-commutative 19.4%

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

      8. hypot-1-def 19.4%

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

   3. Applied egg-rr 19.4%

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

      1. sub-neg 19.4%

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

      2. +-commutative 19.4%

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

      3. metadata-eval 19.4%

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

      4. associate-+l+ 19.4%

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

   5. Applied egg-rr 19.4%

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

   6. Taylor expanded in x around 0 98.6%

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

   if 2 < x

   1. Initial program 48.4%

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

      1. *-un-lft-identity 48.4%

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

      2. *-commutative 48.4%

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

      3. log-prod 48.4%

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

      4. add-sqr-sqrt 48.4%

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

      5. fabs-sqr 48.4%

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

      6. add-sqr-sqrt 48.4%

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

      7. +-commutative 48.4%

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

      8. hypot-1-def 95.8%

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

      9. metadata-eval 95.8%

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

   3. Applied egg-rr 95.8%

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

   4. Taylor expanded in x around inf 97.4%

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

      1. log-rec 97.4%

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

      2. mul-1-neg 97.4%

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

      3. remove-double-neg 97.4%

         \[\leadsto \mathsf{copysign}\left(\left(\log 2 + \color{blue}{\log x}\right) + 0, x\right) \]

   6. Simplified 97.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 2 + \log x\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.1%

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

Alternative 9: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(0.5 \cdot \frac{1}{x} + x \cdot 2\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log1p (+ -1.0 (+ (* 0.5 (/ 1.0 x)) (* 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 <= 1.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(log1pf((-1.0f + ((0.5f * (1.0f / x)) + (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(1.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log1p(Float32(Float32(-1.0) + Float32(Float32(Float32(0.5) * Float32(Float32(1.0) / x)) + Float32(x * Float32(2.0))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. *-commutative 59.9%

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

      3. log-prod 59.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 12.6%

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

      7. +-commutative 12.6%

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

      8. hypot-1-def 12.6%

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

      9. metadata-eval 12.6%

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

   3. Applied egg-rr 12.6%

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

   4. Taylor expanded in x around -inf 97.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)} + 0, x\right) \]

   if -2 < x < 1

   1. Initial program 18.7%

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

      1. log1p-expm1-u 18.7%

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

      2. expm1-udef 18.7%

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

      3. add-exp-log 18.7%

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

      4. add-sqr-sqrt 11.3%

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

      5. fabs-sqr 11.3%

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

      6. add-sqr-sqrt 18.8%

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

      7. +-commutative 18.8%

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

      8. hypot-1-def 18.7%

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

   3. Applied egg-rr 18.7%

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

      1. sub-neg 18.7%

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

      2. +-commutative 18.7%

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

      3. metadata-eval 18.7%

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

      4. associate-+l+ 18.8%

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

   5. Applied egg-rr 18.8%

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

   6. Taylor expanded in x around 0 99.1%

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

   if 1 < x

   1. Initial program 49.2%

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

      1. log1p-expm1-u 49.1%

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

      2. expm1-udef 49.1%

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

      3. add-exp-log 49.2%

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

      4. add-sqr-sqrt 49.1%

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

      5. fabs-sqr 49.1%

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

      6. add-sqr-sqrt 49.2%

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

      7. +-commutative 49.2%

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

      8. hypot-1-def 95.9%

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

   3. Applied egg-rr 95.9%

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

   4. Taylor expanded in x around inf 94.3%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} - 1\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(0.5 \cdot \frac{1}{x} + x \cdot 2\right)\right), x\right)\\ \end{array} \]

Alternative 10: 98.0% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 2.0)
     (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 <= 2.0f) {
		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(2.0))
		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(2.0))
		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 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. *-commutative 59.9%

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

      3. log-prod 59.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 12.6%

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

      7. +-commutative 12.6%

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

      8. hypot-1-def 12.6%

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

      9. metadata-eval 12.6%

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

   3. Applied egg-rr 12.6%

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

   4. Taylor expanded in x around -inf 97.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)} + 0, x\right) \]

   if -2 < x < 2

   1. Initial program 19.4%

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

      1. log1p-expm1-u 19.4%

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

      2. expm1-udef 19.4%

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

      3. add-exp-log 19.4%

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

      4. add-sqr-sqrt 12.0%

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

      5. fabs-sqr 12.0%

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

      6. add-sqr-sqrt 19.4%

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

      7. +-commutative 19.4%

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

      8. hypot-1-def 19.4%

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

   3. Applied egg-rr 19.4%

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

      1. sub-neg 19.4%

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

      2. +-commutative 19.4%

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

      3. metadata-eval 19.4%

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

      4. associate-+l+ 19.4%

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

   5. Applied egg-rr 19.4%

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

   6. Taylor expanded in x around 0 98.6%

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

   if 2 < x

   1. Initial program 48.4%

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

      1. log1p-expm1-u 48.4%

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

      2. expm1-udef 48.4%

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

      3. add-exp-log 48.5%

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

      4. add-sqr-sqrt 48.4%

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

      5. fabs-sqr 48.4%

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

      6. add-sqr-sqrt 48.5%

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

      7. +-commutative 48.5%

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

      8. hypot-1-def 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. sub-neg 95.9%

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

      2. +-commutative 95.9%

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

      3. metadata-eval 95.9%

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

      4. associate-+l+ 95.9%

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

   5. Applied egg-rr 95.9%

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

   6. Taylor expanded in x around inf 97.4%

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

      1. +-commutative 97.4%

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

      2. mul-1-neg 97.4%

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

      3. log-rec 97.4%

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

      4. remove-double-neg 97.4%

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

      5. log-prod 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 97.2%

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

Alternative 11: 84.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 2.0) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(-logf((-1.0f / x)), x);
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(Float32(-log(Float32(Float32(-1.0) / x))), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(-log((single(-1.0) / x)));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 59.3%

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

   2. Taylor expanded in x around -inf 44.1%

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

   if -10 < x < 2

   1. Initial program 20.0%

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

      1. add-sqr-sqrt 19.5%

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

      2. pow2 19.5%

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

      3. log-pow 19.4%

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

      4. add-sqr-sqrt 11.6%

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

      5. fabs-sqr 11.6%

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

      6. add-sqr-sqrt 19.7%

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

      7. +-commutative 19.7%

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

      8. hypot-1-def 19.7%

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

   3. Applied egg-rr 19.7%

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

   4. Taylor expanded in x around 0 16.7%

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

   5. Taylor expanded in x around 0 97.3%

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

   if 2 < x

   1. Initial program 48.4%

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

      1. log1p-expm1-u 48.4%

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

      2. expm1-udef 48.4%

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

      3. add-exp-log 48.5%

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

      4. add-sqr-sqrt 48.4%

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

      5. fabs-sqr 48.4%

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

      6. add-sqr-sqrt 48.5%

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

      7. +-commutative 48.5%

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

      8. hypot-1-def 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. sub-neg 95.9%

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

      2. +-commutative 95.9%

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

      3. metadata-eval 95.9%

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

      4. associate-+l+ 95.9%

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

   5. Applied egg-rr 95.9%

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

   6. Taylor expanded in x around inf 97.4%

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

      1. +-commutative 97.4%

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

      2. mul-1-neg 97.4%

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

      3. log-rec 97.4%

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

      4. remove-double-neg 97.4%

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

      5. log-prod 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 12: 97.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 2.0) (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 <= 2.0f) {
		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(2.0))
		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(2.0))
		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 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. *-commutative 59.9%

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

      3. log-prod 59.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 12.6%

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

      7. +-commutative 12.6%

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

      8. hypot-1-def 12.6%

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

      9. metadata-eval 12.6%

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

   3. Applied egg-rr 12.6%

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

   4. Taylor expanded in x around -inf 97.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)} + 0, x\right) \]

   if -2 < x < 2

   1. Initial program 19.4%

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

      1. add-sqr-sqrt 18.9%

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

      2. pow2 18.9%

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

      3. log-pow 18.8%

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

      4. add-sqr-sqrt 11.7%

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

      5. fabs-sqr 11.7%

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

      6. add-sqr-sqrt 19.1%

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

      7. +-commutative 19.1%

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

      8. hypot-1-def 19.1%

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

   3. Applied egg-rr 19.1%

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

   4. Taylor expanded in x around 0 16.8%

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

   5. Taylor expanded in x around 0 97.9%

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

   if 2 < x

   1. Initial program 48.4%

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

      1. log1p-expm1-u 48.4%

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

      2. expm1-udef 48.4%

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

      3. add-exp-log 48.5%

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

      4. add-sqr-sqrt 48.4%

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

      5. fabs-sqr 48.4%

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

      6. add-sqr-sqrt 48.5%

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

      7. +-commutative 48.5%

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

      8. hypot-1-def 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. sub-neg 95.9%

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

      2. +-commutative 95.9%

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

      3. metadata-eval 95.9%

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

      4. associate-+l+ 95.9%

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

   5. Applied egg-rr 95.9%

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

   6. Taylor expanded in x around inf 97.4%

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

      1. +-commutative 97.4%

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

      2. mul-1-neg 97.4%

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

      3. log-rec 97.4%

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

      4. remove-double-neg 97.4%

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

      5. log-prod 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 96.8%

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

Alternative 13: 76.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= 2.0f) {
		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(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(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 < 2

   1. Initial program 33.0%

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

      1. add-sqr-sqrt 32.7%

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

      2. pow2 32.7%

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

      3. log-pow 32.6%

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

      4. add-sqr-sqrt 7.8%

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

      5. fabs-sqr 7.8%

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

      6. add-sqr-sqrt 16.9%

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

      7. +-commutative 16.9%

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

      8. hypot-1-def 16.9%

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

   3. Applied egg-rr 16.9%

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

   4. Taylor expanded in x around 0 11.2%

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

   5. Taylor expanded in x around 0 68.7%

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

   if 2 < x

   1. Initial program 48.4%

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

      1. log1p-expm1-u 48.4%

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

      2. expm1-udef 48.4%

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

      3. add-exp-log 48.5%

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

      4. add-sqr-sqrt 48.4%

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

      5. fabs-sqr 48.4%

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

      6. add-sqr-sqrt 48.5%

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

      7. +-commutative 48.5%

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

      8. hypot-1-def 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. sub-neg 95.9%

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

      2. +-commutative 95.9%

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

      3. metadata-eval 95.9%

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

      4. associate-+l+ 95.9%

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

   5. Applied egg-rr 95.9%

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

   6. Taylor expanded in x around inf 97.4%

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

      1. +-commutative 97.4%

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

      2. mul-1-neg 97.4%

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

      3. log-rec 97.4%

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

      4. remove-double-neg 97.4%

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

      5. log-prod 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 75.6%

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

Alternative 14: 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 37.2%

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

   1. add-sqr-sqrt 36.9%

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

   2. pow2 36.9%

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

   3. log-pow 36.9%

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

   4. add-sqr-sqrt 18.7%

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

   5. fabs-sqr 18.7%

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

   6. add-sqr-sqrt 25.4%

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

   7. +-commutative 25.4%

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

   8. hypot-1-def 38.1%

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

3. Applied egg-rr 38.1%

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

4. Taylor expanded in x around 0 15.9%

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

5. Taylor expanded in x around 0 53.2%

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

6. Final simplification 53.2%

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

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 2023242 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :herbie-target
  (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))