Rust f32::asinh

Percentage Accurate: 37.7% → 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: 37.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.5)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.4000000059604645)
       (copysign
        (+
         (* -0.16666666666666666 (pow x 3.0))
         (+ (* 0.075 (pow x 5.0)) (+ x (* -0.044642857142857144 (pow x 7.0)))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.5f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.4000000059604645f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + ((0.075f * powf(x, 5.0f)) + (x + (-0.044642857142857144f * powf(x, 7.0f))))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.5))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.4000000059604645))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(Float32(Float32(0.075) * (x ^ Float32(5.0))) + Float32(x + Float32(Float32(-0.044642857142857144) * (x ^ Float32(7.0)))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.5))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.4000000059604645))
		tmp = sign(x) * abs(((single(-0.16666666666666666) * (x ^ single(3.0))) + ((single(0.075) * (x ^ single(5.0))) + (x + (single(-0.044642857142857144) * (x ^ single(7.0)))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -0.5

   1. Initial program 50.3%

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

      1. +-commutative 50.3%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 11.9%

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

      5. flip-+ 8.4%

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

      6. log-div 8.4%

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

      7. add-sqr-sqrt 8.8%

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

      8. fma-def 9.7%

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

      9. +-commutative 9.7%

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

      10. hypot-1-def 9.7%

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

   3. Applied egg-rr 9.7%

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

      1. fma-udef 8.8%

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

      2. +-commutative 8.8%

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

      3. associate--l+ 46.9%

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

      4. +-inverses 97.1%

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

      5. metadata-eval 97.1%

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

      6. metadata-eval 97.1%

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

      7. neg-sub0 97.1%

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

   5. Simplified 97.1%

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

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.400000006

   1. Initial program 23.8%

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

      1. +-commutative 23.8%

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

      2. add-sqr-sqrt 10.1%

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

      3. fabs-sqr 10.1%

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

      4. add-sqr-sqrt 24.3%

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

      5. flip-+ 23.6%

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

      6. add-sqr-sqrt 23.7%

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

      7. fma-def 23.7%

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

      8. +-commutative 23.7%

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

      9. hypot-1-def 23.7%

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

   3. Applied egg-rr 23.7%

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

      1. fma-udef 23.7%

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

      2. +-commutative 23.7%

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

      3. associate--l+ 23.9%

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

      4. +-inverses 23.9%

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

      5. metadata-eval 23.9%

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

   5. Simplified 23.9%

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

   6. Taylor expanded in x around 0 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)}, x\right) \]

   if 0.400000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 63.6%

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

      1. *-un-lft-identity 63.6%

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

      2. *-commutative 63.6%

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

      3. log-prod 63.6%

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

      4. add-sqr-sqrt 63.5%

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

      5. fabs-sqr 63.5%

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

      6. add-sqr-sqrt 63.6%

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

      7. +-commutative 63.6%

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

      8. hypot-1-def 98.5%

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

      9. metadata-eval 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. +-rgt-identity 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.15000000596046448:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.15000000596046448)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.10000000149011612)
       (copysign
        (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.15000000596046448f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - 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(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.15000000596046448))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - 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(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.15000000596046448))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - 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(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -0.150000006

   1. Initial program 50.9%

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

      1. +-commutative 50.9%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 13.2%

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

      5. flip-+ 9.7%

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

      6. log-div 9.8%

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

      7. add-sqr-sqrt 10.1%

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

      8. fma-def 10.9%

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

      9. +-commutative 10.9%

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

      10. hypot-1-def 11.0%

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

   3. Applied egg-rr 11.0%

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

      1. fma-udef 10.1%

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

      2. +-commutative 10.1%

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

      3. associate--l+ 47.6%

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

      4. +-inverses 97.0%

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

      5. metadata-eval 97.0%

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

      6. metadata-eval 97.0%

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

      7. neg-sub0 97.0%

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

   5. Simplified 97.0%

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

   if -0.150000006 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.100000001

   1. Initial program 22.7%

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

      1. +-commutative 22.7%

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

      2. add-sqr-sqrt 9.5%

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

      3. fabs-sqr 9.5%

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

      4. add-sqr-sqrt 23.1%

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

      5. flip-+ 22.4%

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

      6. add-sqr-sqrt 22.6%

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

      7. fma-def 22.6%

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

      8. +-commutative 22.6%

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

      9. hypot-1-def 22.5%

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

   3. Applied egg-rr 22.5%

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

      1. fma-udef 22.5%

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

      2. +-commutative 22.5%

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

      3. associate--l+ 22.8%

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

      4. +-inverses 22.8%

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

      5. metadata-eval 22.8%

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

   5. Simplified 22.8%

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

   6. Taylor expanded in x around 0 99.9%

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

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

   1. Initial program 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. *-commutative 64.0%

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

      3. log-prod 64.0%

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

      4. add-sqr-sqrt 64.0%

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

      5. fabs-sqr 64.0%

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

      6. add-sqr-sqrt 64.0%

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

      7. +-commutative 64.0%

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

      8. hypot-1-def 98.4%

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

      9. metadata-eval 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-rgt-identity 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.15000000596046448:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.10000000149011612))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 10.6%

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

      5. flip-+ 7.1%

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

      6. add-sqr-sqrt 7.3%

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

      7. fma-def 8.3%

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

      8. +-commutative 8.3%

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

      9. hypot-1-def 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. fma-udef 7.3%

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

      2. +-commutative 7.3%

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

      3. associate--l+ 46.0%

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

      4. +-inverses 96.9%

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

      5. metadata-eval 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in x around -inf 97.4%

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

   if -1 < x < 0.100000001

   1. Initial program 23.8%

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

      1. +-commutative 23.8%

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

      2. add-sqr-sqrt 9.4%

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

      3. fabs-sqr 9.4%

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

      4. add-sqr-sqrt 24.3%

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

      5. flip-+ 23.6%

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

      6. add-sqr-sqrt 23.8%

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

      7. fma-def 23.8%

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

      8. +-commutative 23.8%

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

      9. hypot-1-def 23.7%

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

   3. Applied egg-rr 23.7%

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

      1. fma-udef 23.7%

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

      2. +-commutative 23.7%

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

      3. associate--l+ 24.0%

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

      4. +-inverses 24.0%

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

      5. metadata-eval 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in x around 0 98.9%

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

   if 0.100000001 < x

   1. Initial program 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. *-commutative 64.0%

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

      3. log-prod 64.0%

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

      4. add-sqr-sqrt 64.0%

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

      5. fabs-sqr 64.0%

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

      6. add-sqr-sqrt 64.0%

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

      7. +-commutative 64.0%

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

      8. hypot-1-def 98.4%

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

      9. metadata-eval 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-rgt-identity 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.10000000149011612)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.10000000149011612)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.10000000149011612f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.10000000149011612f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.10000000149011612))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.10000000149011612))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.100000001

   1. Initial program 52.1%

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

      1. +-commutative 52.1%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 15.4%

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

      5. flip-+ 12.1%

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

      6. log-div 12.2%

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

      7. add-sqr-sqrt 12.5%

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

      8. fma-def 13.3%

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

      9. +-commutative 13.3%

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

      10. hypot-1-def 13.4%

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

   3. Applied egg-rr 13.4%

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

      1. fma-udef 12.5%

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

      2. +-commutative 12.5%

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

      3. associate--l+ 48.9%

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

      4. +-inverses 96.9%

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

      5. metadata-eval 96.9%

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

      6. metadata-eval 96.9%

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

      7. neg-sub0 96.9%

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

   5. Simplified 96.9%

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

   if -0.100000001 < x < 0.100000001

   1. Initial program 21.5%

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

      1. +-commutative 21.5%

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

      2. add-sqr-sqrt 9.7%

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

      3. fabs-sqr 9.7%

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

      4. add-sqr-sqrt 22.0%

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

      5. flip-+ 21.3%

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

      6. add-sqr-sqrt 21.5%

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

      7. fma-def 21.5%

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

      8. +-commutative 21.5%

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

      9. hypot-1-def 21.4%

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

   3. Applied egg-rr 21.4%

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

      1. fma-udef 21.4%

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

      2. +-commutative 21.4%

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

      3. associate--l+ 21.7%

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

      4. +-inverses 21.7%

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

      5. metadata-eval 21.7%

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

   5. Simplified 21.7%

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

   6. Taylor expanded in x around 0 99.9%

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

   if 0.100000001 < x

   1. Initial program 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. *-commutative 64.0%

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

      3. log-prod 64.0%

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

      4. add-sqr-sqrt 64.0%

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

      5. fabs-sqr 64.0%

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

      6. add-sqr-sqrt 64.0%

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

      7. +-commutative 64.0%

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

      8. hypot-1-def 98.4%

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

      9. metadata-eval 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. +-rgt-identity 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 98.7%

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

Alternative 5: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 10.6%

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

      5. flip-+ 7.1%

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

      6. add-sqr-sqrt 7.3%

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

      7. fma-def 8.3%

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

      8. +-commutative 8.3%

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

      9. hypot-1-def 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. fma-udef 7.3%

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

      2. +-commutative 7.3%

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

      3. associate--l+ 46.0%

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

      4. +-inverses 96.9%

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

      5. metadata-eval 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in x around -inf 97.4%

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

   if -1 < x < 1

   1. Initial program 26.1%

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

      1. +-commutative 26.1%

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

      2. add-sqr-sqrt 12.1%

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

      3. fabs-sqr 12.1%

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

      4. add-sqr-sqrt 26.5%

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

      5. flip-+ 25.8%

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

      6. add-sqr-sqrt 26.0%

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

      7. fma-def 26.0%

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

      8. +-commutative 26.0%

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

      9. hypot-1-def 25.9%

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

   3. Applied egg-rr 25.9%

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

      1. fma-udef 25.9%

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

      2. +-commutative 25.9%

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

      3. associate--l+ 26.2%

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

      4. +-inverses 26.2%

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

      5. metadata-eval 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in x around 0 97.4%

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

   if 1 < x

   1. Initial program 62.0%

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

   2. Taylor expanded in x around inf 96.5%

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

      1. associate-*r/ 96.5%

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

      2. metadata-eval 96.5%

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

      3. rem-square-sqrt 96.5%

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

      4. fabs-sqr 96.5%

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

      5. rem-square-sqrt 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 97.2%

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

Alternative 6: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 10.6%

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

      5. flip-+ 7.1%

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

      6. add-sqr-sqrt 7.3%

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

      7. fma-def 8.3%

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

      8. +-commutative 8.3%

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

      9. hypot-1-def 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. fma-udef 7.3%

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

      2. +-commutative 7.3%

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

      3. associate--l+ 46.0%

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

      4. +-inverses 96.9%

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

      5. metadata-eval 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in x around -inf 97.4%

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

   if -1 < x < 2

   1. Initial program 26.7%

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

      1. +-commutative 26.7%

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

      2. add-sqr-sqrt 12.7%

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

      3. fabs-sqr 12.7%

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

      4. add-sqr-sqrt 27.1%

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

      5. flip-+ 26.4%

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

      6. add-sqr-sqrt 26.6%

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

      7. fma-def 26.6%

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

      8. +-commutative 26.6%

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

      9. hypot-1-def 26.5%

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

   3. Applied egg-rr 26.5%

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

      1. fma-udef 26.5%

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

      2. +-commutative 26.5%

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

      3. associate--l+ 26.7%

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

      4. +-inverses 26.7%

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

      5. metadata-eval 26.7%

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

   5. Simplified 26.7%

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

   6. Taylor expanded in x around 0 96.9%

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

   if 2 < x

   1. Initial program 61.4%

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

   2. Taylor expanded in x around inf 97.3%

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

      1. associate-*r/ 97.3%

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

      2. metadata-eval 97.3%

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

      3. rem-square-sqrt 97.3%

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

      4. fabs-sqr 97.3%

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

      5. rem-square-sqrt 97.3%

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

   4. Simplified 97.3%

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

   5. Taylor expanded in x around 0 96.7%

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

4. Final simplification 97.0%

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

Alternative 7: 97.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 10.6%

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

      5. flip-+ 7.1%

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

      6. add-sqr-sqrt 7.3%

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

      7. fma-def 8.3%

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

      8. +-commutative 8.3%

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

      9. hypot-1-def 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. fma-udef 7.3%

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

      2. +-commutative 7.3%

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

      3. associate--l+ 46.0%

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

      4. +-inverses 96.9%

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

      5. metadata-eval 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in x around -inf 97.4%

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

   if -1 < x < 2

   1. Initial program 26.7%

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

      1. +-commutative 26.7%

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

      2. add-sqr-sqrt 12.7%

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

      3. fabs-sqr 12.7%

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

      4. add-sqr-sqrt 27.1%

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

      5. flip-+ 26.4%

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

      6. add-sqr-sqrt 26.6%

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

      7. fma-def 26.6%

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

      8. +-commutative 26.6%

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

      9. hypot-1-def 26.5%

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

   3. Applied egg-rr 26.5%

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

      1. fma-udef 26.5%

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

      2. +-commutative 26.5%

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

      3. associate--l+ 26.7%

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

      4. +-inverses 26.7%

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

      5. metadata-eval 26.7%

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

   5. Simplified 26.7%

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

   6. Taylor expanded in x around 0 94.9%

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

   if 2 < x

   1. Initial program 61.4%

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

   2. Taylor expanded in x around inf 95.3%

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

      1. rem-square-sqrt 95.3%

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

      2. fabs-sqr 95.3%

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

      3. rem-square-sqrt 95.3%

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

   4. Simplified 95.3%

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

4. Final simplification 95.6%

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

Alternative 8: 97.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 10.6%

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

      5. flip-+ 7.1%

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

      6. add-sqr-sqrt 7.3%

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

      7. fma-def 8.3%

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

      8. +-commutative 8.3%

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

      9. hypot-1-def 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. fma-udef 7.3%

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

      2. +-commutative 7.3%

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

      3. associate--l+ 46.0%

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

      4. +-inverses 96.9%

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

      5. metadata-eval 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in x around -inf 97.4%

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

   if -1 < x < 2

   1. Initial program 26.7%

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

      1. +-commutative 26.7%

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

      2. add-sqr-sqrt 12.7%

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

      3. fabs-sqr 12.7%

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

      4. add-sqr-sqrt 27.1%

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

      5. flip-+ 26.4%

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

      6. add-sqr-sqrt 26.6%

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

      7. fma-def 26.6%

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

      8. +-commutative 26.6%

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

      9. hypot-1-def 26.5%

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

   3. Applied egg-rr 26.5%

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

      1. fma-udef 26.5%

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

      2. +-commutative 26.5%

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

      3. associate--l+ 26.7%

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

      4. +-inverses 26.7%

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

      5. metadata-eval 26.7%

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

   5. Simplified 26.7%

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

   6. Taylor expanded in x around 0 94.9%

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

   if 2 < x

   1. Initial program 61.4%

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

   2. Taylor expanded in x around inf 97.3%

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

      1. associate-*r/ 97.3%

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

      2. metadata-eval 97.3%

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

      3. rem-square-sqrt 97.3%

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

      4. fabs-sqr 97.3%

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

      5. rem-square-sqrt 97.3%

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

   4. Simplified 97.3%

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

   5. Taylor expanded in x around 0 96.7%

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

4. Final simplification 96.0%

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

Alternative 9: 75.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + x)), 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 + 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(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 34.1%

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

      1. +-commutative 34.1%

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

      2. add-sqr-sqrt 8.6%

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

      3. fabs-sqr 8.6%

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

      4. add-sqr-sqrt 21.7%

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

      5. flip-+ 20.1%

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

      6. add-sqr-sqrt 20.3%

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

      7. fma-def 20.6%

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

      8. +-commutative 20.6%

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

      9. hypot-1-def 20.5%

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

   3. Applied egg-rr 20.5%

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

      1. fma-udef 20.2%

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

      2. +-commutative 20.2%

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

      3. associate--l+ 33.0%

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

      4. +-inverses 49.6%

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

      5. metadata-eval 49.6%

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

   5. Simplified 49.6%

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

   6. Taylor expanded in x around 0 67.6%

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

   if 2 < x

   1. Initial program 61.4%

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

   2. Taylor expanded in x around inf 95.3%

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

      1. rem-square-sqrt 95.3%

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

      2. fabs-sqr 95.3%

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

      3. rem-square-sqrt 95.3%

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

   4. Simplified 95.3%

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

4. Final simplification 74.4%

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

Alternative 10: 63.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log 0.125, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0) (copysign (log 0.125) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf(0.125f), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(0.125)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 49.5%

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

   2. Taylor expanded in x around inf -0.0%

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

      1. associate-*r/ -0.0%

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

      2. metadata-eval -0.0%

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

      3. rem-square-sqrt -0.0%

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

      4. fabs-sqr -0.0%

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

      5. rem-square-sqrt -0.0%

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

   4. Simplified -0.0%

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

      1. clear-num -0.0%

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

      2. flip-+ -0.0%

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

      3. frac-add -0.0%

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

      4. *-un-lft-identity -0.0%

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

      5. +-inverses -0.0%

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

      6. +-inverses -0.0%

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

      7. +-inverses -0.0%

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

      8. div-inv -0.0%

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

      9. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{0 + \left(x \cdot \color{blue}{2}\right) \cdot \left(x - x\right)}{\frac{x}{0.5} \cdot \left(x - x\right)}\right), x\right) \]

      10. *-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{0 + \color{blue}{\left(2 \cdot x\right)} \cdot \left(x - x\right)}{\frac{x}{0.5} \cdot \left(x - x\right)}\right), x\right) \]

      11. count-2 -0.0%

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

      12. difference-of-squares -0.0%

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

      13. +-inverses -0.0%

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

      14. metadata-eval -0.0%

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

      15. metadata-eval -0.0%

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

      16. +-inverses -0.0%

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

      17. div-inv -0.0%

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

      18. metadata-eval -0.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{\left(x \cdot x - x \cdot x\right)}^{3}}{\left(x \cdot \color{blue}{2}\right) \cdot \left(x - x\right)}\right), x\right) \]

      19. *-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{{\left(x \cdot x - x \cdot x\right)}^{3}}{\color{blue}{\left(2 \cdot x\right)} \cdot \left(x - x\right)}\right), x\right) \]

      20. count-2 -0.0%

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

      21. difference-of-squares -0.0%

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

      22. +-inverses -0.0%

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

      23. metadata-eval -0.0%

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

      24. +-inverses -0.0%

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

   6. Applied egg-rr 4.0%

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

   8. Simplified 22.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{0.125}, x\right) \]

   if -1 < x

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 28.1%

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

      1. log1p-def 75.2%

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

      2. rem-square-sqrt 39.9%

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

      3. fabs-sqr 39.9%

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

      4. rem-square-sqrt 75.1%

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

   4. Simplified 75.1%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log 0.125, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 11: 15.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(0.8333333333333334, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign 0.8333333333333334 x))

C

float code(float x) {
	return copysignf(0.8333333333333334f, x);
}

Julia

function code(x)
	return copysign(Float32(0.8333333333333334), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(0.8333333333333334));
end

Derivation

1. Initial program 40.8%

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

   1. +-commutative 40.8%

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

   2. add-sqr-sqrt 21.6%

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

   3. fabs-sqr 21.6%

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

   4. add-sqr-sqrt 31.5%

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

   5. flip-+ 17.7%

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

   6. add-sqr-sqrt 17.8%

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

   7. fma-def 18.0%

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

   8. +-commutative 18.0%

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

   9. hypot-1-def 18.0%

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

3. Applied egg-rr 18.0%

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

   1. fma-udef 17.7%

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

   2. +-commutative 17.7%

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

   3. associate--l+ 28.1%

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

   4. +-inverses 41.3%

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

   5. metadata-eval 41.3%

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

5. Simplified 41.3%

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

6. Taylor expanded in x around 0 53.4%

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

8. Simplified 16.1%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{0.8333333333333334}, x\right) \]

9. Final simplification 16.1%

   \[\leadsto \mathsf{copysign}\left(0.8333333333333334, x\right) \]

Alternative 12: 16.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(1.09375, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign 1.09375 x))

C

float code(float x) {
	return copysignf(1.09375f, x);
}

Julia

function code(x)
	return copysign(Float32(1.09375), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(1.09375));
end

Derivation

1. Initial program 40.8%

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

   1. +-commutative 40.8%

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

   2. add-sqr-sqrt 21.6%

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

   3. fabs-sqr 21.6%

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

   4. add-sqr-sqrt 31.5%

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

   5. flip-+ 17.7%

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

   6. add-sqr-sqrt 17.8%

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

   7. fma-def 18.0%

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

   8. +-commutative 18.0%

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

   9. hypot-1-def 18.0%

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

3. Applied egg-rr 18.0%

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

   1. fma-udef 17.7%

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

   2. +-commutative 17.7%

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

   3. associate--l+ 28.1%

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

   4. +-inverses 41.3%

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

   5. metadata-eval 41.3%

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

5. Simplified 41.3%

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

6. Taylor expanded in x around -inf 25.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log \left(\frac{-1}{x}\right) + \left(\log 0.5 + 0.09375 \cdot \frac{1}{{x}^{4}}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}}, x\right) \]

8. Simplified 16.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{1.09375}, x\right) \]

9. Final simplification 16.3%

   \[\leadsto \mathsf{copysign}\left(1.09375, x\right) \]

Alternative 13: 16.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(1.9583333333333333, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign 1.9583333333333333 x))

C

float code(float x) {
	return copysignf(1.9583333333333333f, x);
}

Julia

function code(x)
	return copysign(Float32(1.9583333333333333), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(1.9583333333333333));
end

Derivation

1. Initial program 40.8%

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

   1. +-commutative 40.8%

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

   2. add-sqr-sqrt 21.6%

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

   3. fabs-sqr 21.6%

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

   4. add-sqr-sqrt 31.5%

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

   5. flip-+ 17.7%

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

   6. add-sqr-sqrt 17.8%

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

   7. fma-def 18.0%

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

   8. +-commutative 18.0%

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

   9. hypot-1-def 18.0%

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

3. Applied egg-rr 18.0%

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

   1. fma-udef 17.7%

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

   2. +-commutative 17.7%

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

   3. associate--l+ 28.1%

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

   4. +-inverses 41.3%

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

   5. metadata-eval 41.3%

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

5. Simplified 41.3%

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

6. Taylor expanded in x around inf 25.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\log 2 + \left(0.052083333333333336 \cdot \frac{1}{{x}^{6}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)\right) - 0.09375 \cdot \frac{1}{{x}^{4}}}, x\right) \]

8. Simplified 16.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{1.9583333333333333}, x\right) \]

9. Final simplification 16.7%

   \[\leadsto \mathsf{copysign}\left(1.9583333333333333, x\right) \]

Alternative 14: 53.7% 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 40.8%

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

   1. +-commutative 40.8%

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

   2. add-sqr-sqrt 21.6%

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

   3. fabs-sqr 21.6%

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

   4. add-sqr-sqrt 31.5%

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

   5. flip-+ 17.7%

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

   6. add-sqr-sqrt 17.8%

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

   7. fma-def 18.0%

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

   8. +-commutative 18.0%

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

   9. hypot-1-def 18.0%

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

3. Applied egg-rr 18.0%

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

   1. fma-udef 17.7%

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

   2. +-commutative 17.7%

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

   3. associate--l+ 28.1%

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

   4. +-inverses 41.3%

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

   5. metadata-eval 41.3%

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

5. Simplified 41.3%

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

6. Taylor expanded in x around 0 54.0%

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

7. Final simplification 54.0%

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

Developer target: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2023193 
(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))