Rust f32::asinh

Percentage Accurate: 38.2% → 99.0%

Time: 7.5s

Alternatives: 14

Speedup: 3.8×

• Unsound rule application detected in e-graph. Results may not be sound.

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 38.2% 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: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := {\left(x + 1\right)}^{2}\\ \mathbf{if}\;t_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664 \cdot \left(\frac{3}{x + 1} + \frac{3}{t_1}\right), {x}^{4}, \mathsf{fma}\left(0.001388888888888889 \cdot \left(\frac{45}{t_1} + \left(\frac{45}{x + 1} + \frac{30}{{\left(x + 1\right)}^{3}}\right)\right), {x}^{6}, \mathsf{fma}\left(0.5, \frac{x \cdot x}{x + 1}, \mathsf{log1p}\left(x\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (pow (+ x 1.0) 2.0)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
     (if (<= t_0 1.0)
       (copysign
        (fma
         (* -0.041666666666666664 (+ (/ 3.0 (+ x 1.0)) (/ 3.0 t_1)))
         (pow x 4.0)
         (fma
          (*
           0.001388888888888889
           (+
            (/ 45.0 t_1)
            (+ (/ 45.0 (+ x 1.0)) (/ 30.0 (pow (+ x 1.0) 3.0)))))
          (pow x 6.0)
          (fma 0.5 (/ (* x x) (+ x 1.0)) (log1p x))))
        x)
       (copysign (log (/ 0.5 x)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = powf((x + 1.0f), 2.0f);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (t_0 <= 1.0f) {
		tmp = copysignf(fmaf((-0.041666666666666664f * ((3.0f / (x + 1.0f)) + (3.0f / t_1))), powf(x, 4.0f), fmaf((0.001388888888888889f * ((45.0f / t_1) + ((45.0f / (x + 1.0f)) + (30.0f / powf((x + 1.0f), 3.0f))))), powf(x, 6.0f), fmaf(0.5f, ((x * x) / (x + 1.0f)), log1pf(x)))), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = Float32(x + Float32(1.0)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(1.0))
		tmp = copysign(fma(Float32(Float32(-0.041666666666666664) * Float32(Float32(Float32(3.0) / Float32(x + Float32(1.0))) + Float32(Float32(3.0) / t_1))), (x ^ Float32(4.0)), fma(Float32(Float32(0.001388888888888889) * Float32(Float32(Float32(45.0) / t_1) + Float32(Float32(Float32(45.0) / Float32(x + Float32(1.0))) + Float32(Float32(30.0) / (Float32(x + Float32(1.0)) ^ Float32(3.0)))))), (x ^ Float32(6.0)), fma(Float32(0.5), Float32(Float32(x * x) / Float32(x + Float32(1.0))), log1p(x)))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 58.1%

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

      1. +-commutative 58.1%

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

      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 13.4%

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

      5. flip-+ 10.5%

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

      6. add-sqr-sqrt 12.1%

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

      7. fma-def 12.1%

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

      8. +-commutative 12.1%

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

      9. hypot-1-def 12.1%

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

   3. Applied egg-rr 12.1%

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

      1. fma-udef 12.1%

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

      2. +-commutative 12.1%

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

      3. associate--l+ 55.3%

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

      4. +-inverses 98.5%

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

      5. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 1

   1. Initial program 23.7%

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

   2. Taylor expanded in x around 0 23.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.041666666666666664 \cdot \left(\left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}\right) + \left(0.001388888888888889 \cdot \left(\left(45 \cdot \frac{1}{1 + \left|x\right|} + \left(30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}} + 45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) \cdot {x}^{6}\right) + \left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right)\right)}, x\right) \]
   3. Step-by-step derivation

      1. associate-*r* 23.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-0.041666666666666664 \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) \cdot {x}^{4}} + \left(0.001388888888888889 \cdot \left(\left(45 \cdot \frac{1}{1 + \left|x\right|} + \left(30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}} + 45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) \cdot {x}^{6}\right) + \left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right)\right), x\right) \]

      2. fma-def 23.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(-0.041666666666666664 \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right), {x}^{4}, 0.001388888888888889 \cdot \left(\left(45 \cdot \frac{1}{1 + \left|x\right|} + \left(30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}} + 45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) \cdot {x}^{6}\right) + \left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right)\right)}, x\right) \]

   4. Simplified 99.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(-0.041666666666666664 \cdot \left(\frac{3}{1 + x} + \frac{3}{{\left(1 + x\right)}^{2}}\right), {x}^{4}, \mathsf{fma}\left(0.001388888888888889 \cdot \left(\frac{45}{{\left(1 + x\right)}^{2}} + \left(\frac{45}{1 + x} + \frac{30}{{\left(1 + x\right)}^{3}}\right)\right), {x}^{6}, \mathsf{fma}\left(0.5, \frac{x \cdot x}{1 + x}, \mathsf{log1p}\left(x\right)\right)\right)\right)}, x\right) \]

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

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

      3. rem-square-sqrt 95.0%

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

      4. fabs-sqr 95.0%

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

      5. rem-square-sqrt 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in x around 0 97.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664 \cdot \left(\frac{3}{x + 1} + \frac{3}{{\left(x + 1\right)}^{2}}\right), {x}^{4}, \mathsf{fma}\left(0.001388888888888889 \cdot \left(\frac{45}{{\left(x + 1\right)}^{2}} + \left(\frac{45}{x + 1} + \frac{30}{{\left(x + 1\right)}^{3}}\right)\right), {x}^{6}, \mathsf{fma}\left(0.5, \frac{x \cdot x}{x + 1}, \mathsf{log1p}\left(x\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
     (if (<= t_0 1.0)
       (copysign
        (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
        x)
       (copysign (log (/ 0.5 x)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (t_0 <= 1.0f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + (x + (0.075f * powf(x, 5.0f)))), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(1.0))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(x + Float32(Float32(0.075) * (x ^ Float32(5.0))))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.20000000298023224))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - x))));
	elseif (t_0 <= single(1.0))
		tmp = sign(x) * abs(((single(-0.16666666666666666) * (x ^ single(3.0))) + (x + (single(0.075) * (x ^ single(5.0))))));
	else
		tmp = sign(x) * abs(log((single(0.5) / 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) < -0.200000003

   1. Initial program 58.1%

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

      1. +-commutative 58.1%

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

      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 13.4%

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

      5. flip-+ 10.5%

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

      6. add-sqr-sqrt 12.1%

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

      7. fma-def 12.1%

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

      8. +-commutative 12.1%

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

      9. hypot-1-def 12.1%

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

   3. Applied egg-rr 12.1%

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

      1. fma-udef 12.1%

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

      2. +-commutative 12.1%

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

      3. associate--l+ 55.3%

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

      4. +-inverses 98.5%

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

      5. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 1

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. add-sqr-sqrt 12.4%

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

      3. fabs-sqr 12.4%

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

      4. add-sqr-sqrt 23.8%

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

      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 23.5%

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

      7. fma-def 23.5%

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

      8. +-commutative 23.5%

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

      9. hypot-1-def 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+ 23.8%

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

      4. +-inverses 23.8%

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

      5. metadata-eval 23.8%

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

   5. Simplified 23.8%

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

   6. Taylor expanded in x around 0 99.4%

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

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

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

      3. rem-square-sqrt 95.0%

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

      4. fabs-sqr 95.0%

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

      5. rem-square-sqrt 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in x around 0 97.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05000000074505806)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.05000000074505806f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - 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);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05000000074505806))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - 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(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.05000000074505806))
		tmp = sign(x) * abs(log((single(1.0) / (hypot(single(1.0), x) - 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)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 58.6%

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

      1. +-commutative 58.6%

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

      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 14.6%

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

      5. flip-+ 11.7%

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

      6. add-sqr-sqrt 13.4%

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

      7. fma-def 13.4%

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

      8. +-commutative 13.4%

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

      9. hypot-1-def 13.4%

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

   3. Applied egg-rr 13.4%

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

      1. fma-udef 13.4%

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

      2. +-commutative 13.4%

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

      3. associate--l+ 55.9%

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

      4. +-inverses 98.4%

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

      5. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   if -0.0500000007 < x < 1

   1. Initial program 23.2%

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

      1. +-commutative 23.2%

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

      2. add-sqr-sqrt 12.5%

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

      3. fabs-sqr 12.5%

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

      4. add-sqr-sqrt 23.2%

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

      5. flip-+ 23.0%

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

      6. add-sqr-sqrt 23.0%

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

      7. fma-def 23.0%

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

      8. +-commutative 23.0%

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

      9. hypot-1-def 23.1%

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

   3. Applied egg-rr 23.1%

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

      1. fma-udef 23.1%

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

      2. +-commutative 23.1%

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

      3. associate--l+ 23.2%

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

      4. +-inverses 23.2%

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

      5. metadata-eval 23.2%

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

   5. Simplified 23.2%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

      3. rem-square-sqrt 95.0%

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

      4. fabs-sqr 95.0%

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

      5. rem-square-sqrt 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in x around 0 97.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 4: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05000000074505806)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.05000000074505806f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - 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);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05000000074505806))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(1.0))
		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(-0.05000000074505806))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - 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)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 58.6%

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

      1. +-commutative 58.6%

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

      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 14.6%

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

      5. flip-+ 11.7%

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

      6. log-div 11.7%

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

      7. add-sqr-sqrt 13.3%

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

      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 13.4%

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

      2. +-commutative 13.4%

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

      3. associate--l+ 55.8%

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

      4. +-inverses 98.3%

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

      5. metadata-eval 98.3%

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

      6. metadata-eval 98.3%

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

      7. neg-sub0 98.3%

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

   5. Simplified 98.3%

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

   if -0.0500000007 < x < 1

   1. Initial program 23.2%

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

      1. +-commutative 23.2%

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

      2. add-sqr-sqrt 12.5%

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

      3. fabs-sqr 12.5%

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

      4. add-sqr-sqrt 23.2%

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

      5. flip-+ 23.0%

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

      6. add-sqr-sqrt 23.0%

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

      7. fma-def 23.0%

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

      8. +-commutative 23.0%

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

      9. hypot-1-def 23.1%

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

   3. Applied egg-rr 23.1%

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

      1. fma-udef 23.1%

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

      2. +-commutative 23.1%

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

      3. associate--l+ 23.2%

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

      4. +-inverses 23.2%

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

      5. metadata-eval 23.2%

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

   5. Simplified 23.2%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

      3. rem-square-sqrt 95.0%

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

      4. fabs-sqr 95.0%

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

      5. rem-square-sqrt 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in x around 0 97.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 5: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\frac{-0.5}{x} + x \cdot -2}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ 1.0 (+ (/ -0.5 x) (* x -2.0)))) x)
   (if (<= x 1.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((1.0f / ((-0.5f / x) + (x * -2.0f)))), x);
	} else if (x <= 1.0f) {
		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(-10.0))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(Float32(Float32(-0.5) / x) + Float32(x * Float32(-2.0))))), 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(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(1.0) / ((single(-0.5) / x) + (x * single(-2.0))))));
	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)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.8%

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

      1. +-commutative 56.8%

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

      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 10.8%

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

      5. flip-+ 7.7%

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

      6. add-sqr-sqrt 9.4%

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

      7. fma-def 9.4%

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

      8. +-commutative 9.4%

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

      9. hypot-1-def 9.4%

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

   3. Applied egg-rr 9.4%

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

      1. fma-udef 9.4%

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

      2. +-commutative 9.4%

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

      3. associate--l+ 53.9%

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

      4. +-inverses 98.5%

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

      5. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around -inf 98.4%

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

      1. cancel-sign-sub-inv 98.4%

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

      2. metadata-eval 98.4%

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

      3. div-inv 98.4%

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

      4. +-commutative 98.4%

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

      5. *-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

   if -10 < x < 1

   1. Initial program 24.9%

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

      1. +-commutative 24.9%

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

      2. add-sqr-sqrt 12.2%

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

      3. fabs-sqr 12.2%

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

      4. add-sqr-sqrt 25.0%

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

      5. flip-+ 24.8%

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

      6. add-sqr-sqrt 24.7%

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

      7. fma-def 24.7%

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

      8. +-commutative 24.7%

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

      9. hypot-1-def 24.9%

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

   3. Applied egg-rr 24.9%

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

      1. fma-udef 24.9%

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

      2. +-commutative 24.9%

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

      3. associate--l+ 25.0%

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

      4. +-inverses 25.0%

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

      5. metadata-eval 25.0%

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

   5. Simplified 25.0%

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

   6. Taylor expanded in x around 0 98.3%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

      3. rem-square-sqrt 95.0%

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

      4. fabs-sqr 95.0%

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

      5. rem-square-sqrt 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in x around 0 97.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\frac{-0.5}{x} + x \cdot -2}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 6: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (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))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		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);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		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(0.5) / x)), x);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.8%

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

      1. +-commutative 56.8%

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

      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 10.8%

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

      5. flip-+ 7.7%

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

      6. add-sqr-sqrt 9.4%

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

      7. fma-def 9.4%

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

      8. +-commutative 9.4%

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

      9. hypot-1-def 9.4%

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

   3. Applied egg-rr 9.4%

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

      1. fma-udef 9.4%

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

      2. +-commutative 9.4%

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

      3. associate--l+ 53.9%

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

      4. +-inverses 98.5%

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

      5. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around -inf 98.2%

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

   if -10 < x < 1

   1. Initial program 24.9%

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

      1. +-commutative 24.9%

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

      2. add-sqr-sqrt 12.2%

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

      3. fabs-sqr 12.2%

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

      4. add-sqr-sqrt 25.0%

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

      5. flip-+ 24.8%

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

      6. add-sqr-sqrt 24.7%

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

      7. fma-def 24.7%

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

      8. +-commutative 24.7%

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

      9. hypot-1-def 24.9%

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

   3. Applied egg-rr 24.9%

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

      1. fma-udef 24.9%

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

      2. +-commutative 24.9%

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

      3. associate--l+ 25.0%

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

      4. +-inverses 25.0%

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

      5. metadata-eval 25.0%

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

   5. Simplified 25.0%

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

   6. Taylor expanded in x around 0 98.3%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

      3. rem-square-sqrt 95.0%

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

      4. fabs-sqr 95.0%

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

      5. rem-square-sqrt 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in x around 0 97.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 7: 96.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(x * Float32(-2.0))), x);
	elseif (x <= Float32(1.0))
		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(-10.0))
		tmp = sign(x) * abs(log((x * single(-2.0))));
	elseif (x <= single(1.0))
		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 < -10

   1. Initial program 56.8%

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

      1. +-commutative 56.8%

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

      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 10.8%

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

      5. flip-+ 7.7%

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

      6. add-sqr-sqrt 9.4%

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

      7. fma-def 9.4%

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

      8. +-commutative 9.4%

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

      9. hypot-1-def 9.4%

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

   3. Applied egg-rr 9.4%

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

      1. fma-udef 9.4%

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

      2. +-commutative 9.4%

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

      3. associate--l+ 53.9%

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

      4. +-inverses 98.5%

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

      5. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around -inf 98.4%

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

   7. Taylor expanded in x around 0 96.8%

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

      1. *-commutative 96.8%

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

   9. Simplified 96.8%

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

   if -10 < x < 1

   1. Initial program 24.9%

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

      1. +-commutative 24.9%

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

      2. add-sqr-sqrt 12.2%

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

      3. fabs-sqr 12.2%

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

      4. add-sqr-sqrt 25.0%

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

      5. flip-+ 24.8%

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

      6. add-sqr-sqrt 24.7%

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

      7. fma-def 24.7%

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

      8. +-commutative 24.7%

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

      9. hypot-1-def 24.9%

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

   3. Applied egg-rr 24.9%

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

      1. fma-udef 24.9%

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

      2. +-commutative 24.9%

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

      3. associate--l+ 25.0%

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

      4. +-inverses 25.0%

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

      5. metadata-eval 25.0%

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

   5. Simplified 25.0%

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

   6. Taylor expanded in x around 0 96.5%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 93.9%

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

      1. rem-square-sqrt 93.8%

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

      2. fabs-sqr 93.8%

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

      3. rem-square-sqrt 93.9%

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

   4. Simplified 93.9%

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

4. Final simplification 95.9%

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

Alternative 8: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		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(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		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 < -10

   1. Initial program 56.8%

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

      1. +-commutative 56.8%

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

      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 10.8%

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

      5. flip-+ 7.7%

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

      6. add-sqr-sqrt 9.4%

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

      7. fma-def 9.4%

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

      8. +-commutative 9.4%

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

      9. hypot-1-def 9.4%

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

   3. Applied egg-rr 9.4%

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

      1. fma-udef 9.4%

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

      2. +-commutative 9.4%

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

      3. associate--l+ 53.9%

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

      4. +-inverses 98.5%

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

      5. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around -inf 98.2%

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

   if -10 < x < 1

   1. Initial program 24.9%

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

      1. +-commutative 24.9%

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

      2. add-sqr-sqrt 12.2%

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

      3. fabs-sqr 12.2%

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

      4. add-sqr-sqrt 25.0%

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

      5. flip-+ 24.8%

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

      6. add-sqr-sqrt 24.7%

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

      7. fma-def 24.7%

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

      8. +-commutative 24.7%

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

      9. hypot-1-def 24.9%

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

   3. Applied egg-rr 24.9%

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

      1. fma-udef 24.9%

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

      2. +-commutative 24.9%

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

      3. associate--l+ 25.0%

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

      4. +-inverses 25.0%

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

      5. metadata-eval 25.0%

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

   5. Simplified 25.0%

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

   6. Taylor expanded in x around 0 96.5%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 93.9%

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

      1. rem-square-sqrt 93.8%

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

      2. fabs-sqr 93.8%

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

      3. rem-square-sqrt 93.9%

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

   4. Simplified 93.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 9: 97.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		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(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		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(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		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 < -10

   1. Initial program 56.8%

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

      1. +-commutative 56.8%

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

      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 10.8%

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

      5. flip-+ 7.7%

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

      6. add-sqr-sqrt 9.4%

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

      7. fma-def 9.4%

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

      8. +-commutative 9.4%

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

      9. hypot-1-def 9.4%

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

   3. Applied egg-rr 9.4%

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

      1. fma-udef 9.4%

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

      2. +-commutative 9.4%

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

      3. associate--l+ 53.9%

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

      4. +-inverses 98.5%

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

      5. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around -inf 98.2%

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

   if -10 < x < 1

   1. Initial program 24.9%

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

      1. +-commutative 24.9%

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

      2. add-sqr-sqrt 12.2%

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

      3. fabs-sqr 12.2%

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

      4. add-sqr-sqrt 25.0%

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

      5. flip-+ 24.8%

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

      6. add-sqr-sqrt 24.7%

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

      7. fma-def 24.7%

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

      8. +-commutative 24.7%

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

      9. hypot-1-def 24.9%

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

   3. Applied egg-rr 24.9%

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

      1. fma-udef 24.9%

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

      2. +-commutative 24.9%

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

      3. associate--l+ 25.0%

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

      4. +-inverses 25.0%

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

      5. metadata-eval 25.0%

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

   5. Simplified 25.0%

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

   6. Taylor expanded in x around 0 96.5%

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

   if 1 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. associate-*r/ 95.1%

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

      2. metadata-eval 95.1%

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

      3. rem-square-sqrt 95.0%

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

      4. fabs-sqr 95.0%

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

      5. rem-square-sqrt 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in x around 0 97.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 10: 81.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0) (copysign (log (* x -2.0)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((x * -2.0f)), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(x * Float32(-2.0))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -10

   1. Initial program 56.8%

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

      1. +-commutative 56.8%

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

      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 10.8%

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

      5. flip-+ 7.7%

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

      6. add-sqr-sqrt 9.4%

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

      7. fma-def 9.4%

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

      8. +-commutative 9.4%

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

      9. hypot-1-def 9.4%

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

   3. Applied egg-rr 9.4%

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

      1. fma-udef 9.4%

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

      2. +-commutative 9.4%

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

      3. associate--l+ 53.9%

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

      4. +-inverses 98.5%

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

      5. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around -inf 98.4%

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

   7. Taylor expanded in x around 0 96.8%

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

      1. *-commutative 96.8%

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

   9. Simplified 96.8%

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

   if -10 < x

   1. Initial program 35.3%

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

   2. Taylor expanded in x around 0 28.9%

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

      1. log1p-def 75.1%

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

      2. rem-square-sqrt 46.5%

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

      3. fabs-sqr 46.5%

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

      4. rem-square-sqrt 75.0%

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

   4. Simplified 75.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 11: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0) (copysign 15.333333333333334 x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(15.333333333333334f, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(Float32(15.333333333333334), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -10

   1. Initial program 56.8%

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

   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. Taylor expanded in x around 0 -0.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{4 \cdot {x}^{2} + \left(\log 0.5 + \left(-8 \cdot {x}^{4} + \left(21.333333333333332 \cdot {x}^{6} + -1 \cdot \log x\right)\right)\right)}, x\right) \]

   7. Simplified 27.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{15.333333333333334}, x\right) \]

   if -10 < x

   1. Initial program 35.3%

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

   2. Taylor expanded in x around 0 28.9%

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

      1. log1p-def 75.1%

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

      2. rem-square-sqrt 46.5%

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

      3. fabs-sqr 46.5%

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

      4. rem-square-sqrt 75.0%

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

   4. Simplified 75.0%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 12: 62.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{elif}\;x \leq 10:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign 15.333333333333334 x)
   (if (<= x 10.0) (copysign x x) (copysign 15.333333333333334 x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(15.333333333333334f, x);
	} else if (x <= 10.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(15.333333333333334f, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(Float32(15.333333333333334), x);
	elseif (x <= Float32(10.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(15.333333333333334), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(single(15.333333333333334));
	elseif (x <= single(10.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(single(15.333333333333334));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -10 or 10 < x

   1. Initial program 54.6%

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

   2. Taylor expanded in x around inf 49.4%

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

      1. associate-*r/ 49.4%

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

      2. metadata-eval 49.4%

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

      3. rem-square-sqrt 49.3%

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

      4. fabs-sqr 49.3%

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

      5. rem-square-sqrt 49.4%

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

   4. Simplified 49.4%

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

   5. Taylor expanded in x around 0 1.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{4 \cdot {x}^{2} + \left(\log 0.5 + \left(-8 \cdot {x}^{4} + \left(21.333333333333332 \cdot {x}^{6} + -1 \cdot \log x\right)\right)\right)}, x\right) \]

   7. Simplified 27.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{15.333333333333334}, x\right) \]

   if -10 < x < 10

   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 13.6%

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

      3. fabs-sqr 13.6%

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

      4. add-sqr-sqrt 26.2%

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

      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 25.8%

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

      7. fma-def 25.8%

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

      8. +-commutative 25.8%

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

      9. hypot-1-def 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+ 26.0%

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

      4. +-inverses 26.0%

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

      5. metadata-eval 26.0%

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

   5. Simplified 26.0%

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

   6. Taylor expanded in x around 0 95.4%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{elif}\;x \leq 10:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \end{array} \]

Alternative 13: 17.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(-6, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign -6.0 x))

C

float code(float x) {
	return copysignf(-6.0f, x);
}

Julia

function code(x)
	return copysign(Float32(-6.0), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(-6.0));
end

Derivation

1. Initial program 40.6%

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

2. Taylor expanded in x around inf 28.2%

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

   1. associate-*r/ 28.2%

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

   2. metadata-eval 28.2%

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

   3. rem-square-sqrt 28.1%

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

   4. fabs-sqr 28.1%

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

   5. rem-square-sqrt 28.2%

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

4. Simplified 28.2%

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

5. Taylor expanded in x around 0 3.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{4 \cdot {x}^{2} + \left(\log 0.5 + \left(-8 \cdot {x}^{4} + -1 \cdot \log x\right)\right)}, x\right) \]

7. Simplified 17.8%

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

8. Final simplification 17.8%

   \[\leadsto \mathsf{copysign}\left(-6, x\right) \]

Alternative 14: 18.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(15.333333333333334, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign 15.333333333333334 x))

C

float code(float x) {
	return copysignf(15.333333333333334f, x);
}

Julia

function code(x)
	return copysign(Float32(15.333333333333334), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(single(15.333333333333334));
end

Derivation

1. Initial program 40.6%

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

2. Taylor expanded in x around inf 28.2%

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

   1. associate-*r/ 28.2%

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

   2. metadata-eval 28.2%

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

   3. rem-square-sqrt 28.1%

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

   4. fabs-sqr 28.1%

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

   5. rem-square-sqrt 28.2%

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

4. Simplified 28.2%

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

5. Taylor expanded in x around 0 3.1%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{4 \cdot {x}^{2} + \left(\log 0.5 + \left(-8 \cdot {x}^{4} + \left(21.333333333333332 \cdot {x}^{6} + -1 \cdot \log x\right)\right)\right)}, x\right) \]

7. Simplified 19.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{15.333333333333334}, x\right) \]

8. Final simplification 19.2%

   \[\leadsto \mathsf{copysign}\left(15.333333333333334, x\right) \]

Developer target: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2023174 
(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))