Rust f32::asinh

Percentage Accurate: 37.6% → 98.4%

Time: 8.6s

Alternatives: 11

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -4:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -4.0)
   (copysign (log (- (/ 0.125 (pow x 3.0)) (/ 0.5 x))) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= -4.0f) {
		tmp = copysignf(logf(((0.125f / powf(x, 3.0f)) - (0.5f / x))), x);
	} else {
		tmp = copysignf(log1pf((x + (hypotf(1.0f, x) + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x) <= Float32(-4.0))
		tmp = copysign(log(Float32(Float32(Float32(0.125) / (x ^ Float32(3.0))) - Float32(Float32(0.5) / x))), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(hypot(Float32(1.0), x) + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -4

   1. Initial program 46.8%

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

      1. +-commutative 46.8%

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

      2. hypot-1-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-lft-identity 98.6%

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

      2. +-commutative 98.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. fabs-sqr -0.0%

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

      5. add-sqr-sqrt 10.5%

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

   5. Applied egg-rr 10.5%

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

   6. Taylor expanded in x around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   if -4 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 32.5%

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

      1. +-commutative 32.5%

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

      2. hypot-1-def 48.3%

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

   3. Simplified 48.3%

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

      1. log1p-expm1-u 48.3%

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

      2. expm1-udef 48.3%

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

      3. add-exp-log 48.3%

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

      4. associate--l+ 97.7%

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

      5. add-sqr-sqrt 63.1%

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

      6. fabs-sqr 63.1%

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

      7. add-sqr-sqrt 97.6%

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

   5. Applied egg-rr 97.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -4:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 2: 98.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (/ -0.5 x)) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf(log1pf((x + (hypotf(1.0f, x) + -1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-50.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(hypot(Float32(1.0), x) + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -50

   1. Initial program 45.1%

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

      1. +-commutative 45.1%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. +-lft-identity 98.5%

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

      2. +-commutative 98.5%

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

      3. add-sqr-sqrt -0.0%

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

      4. fabs-sqr -0.0%

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

      5. add-sqr-sqrt 8.5%

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

   5. Applied egg-rr 8.5%

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

   6. Taylor expanded in x around -inf 99.4%

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

   if -50 < x

   1. Initial program 33.2%

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

      1. +-commutative 33.2%

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

      2. hypot-1-def 48.9%

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

   3. Simplified 48.9%

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

      1. log1p-expm1-u 48.9%

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

      2. expm1-udef 48.9%

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

      3. add-exp-log 48.9%

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

      4. associate--l+ 97.7%

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

      5. add-sqr-sqrt 62.4%

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

      6. fabs-sqr 62.4%

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

      7. add-sqr-sqrt 97.3%

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

   5. Applied egg-rr 97.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 3: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.05000000074505806)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf(((x * 2.0f) + (0.5f * (1.0f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(x * Float32(2.0)) + Float32(Float32(0.5) * Float32(Float32(1.0) / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.05000000074505806))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log(((x * single(2.0)) + (single(0.5) * (single(1.0) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 48.3%

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

      1. +-commutative 48.3%

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

      2. hypot-1-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-lft-identity 98.6%

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

      2. +-commutative 98.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. fabs-sqr -0.0%

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

      5. add-sqr-sqrt 13.0%

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

   5. Applied egg-rr 13.0%

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

   6. Taylor expanded in x around -inf 97.0%

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

   if -1 < x < 0.0500000007

   1. Initial program 22.8%

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

      1. +-commutative 22.8%

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

      2. hypot-1-def 23.0%

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

   3. Simplified 23.0%

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

   4. Taylor expanded in x around 0 21.9%

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

      1. *-un-lft-identity 21.9%

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

      2. log-prod 21.9%

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

      3. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} + \log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. +-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right) + 0}, x\right) \]

      5. log1p-def 98.6%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + 0.5 \cdot {x}^{2}\right)} + 0, x\right) \]

      6. +-commutative 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot {x}^{2} + \left|x\right|}\right) + 0, x\right) \]

      7. fma-def 98.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)}\right) + 0, x\right) \]

      8. add-sqr-sqrt 48.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right) + 0, x\right) \]

      9. fabs-sqr 48.2%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right) + 0, x\right) \]

      10. add-sqr-sqrt 98.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right)\right) + 0, x\right) \]

   6. Applied egg-rr 98.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 98.4%

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

      2. fma-udef 98.4%

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

      3. *-commutative 98.4%

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

      4. fma-udef 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in x around 0 98.8%

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

   if 0.0500000007 < x

   1. Initial program 50.9%

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

      1. +-commutative 50.9%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. +-lft-identity 99.9%

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

      2. +-commutative 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. fabs-sqr 99.9%

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

      5. add-sqr-sqrt 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 96.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \]

Alternative 4: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.5)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.5f) {
		tmp = copysignf((x + (powf(x, 3.0f) * -0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.5))
		tmp = copysign(Float32(x + Float32((x ^ Float32(3.0)) * Float32(-0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.5))
		tmp = sign(x) * abs((x + ((x ^ single(3.0)) * single(-0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 48.3%

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

      1. +-commutative 48.3%

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

      2. hypot-1-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-lft-identity 98.6%

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

      2. +-commutative 98.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. fabs-sqr -0.0%

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

      5. add-sqr-sqrt 13.0%

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

   5. Applied egg-rr 13.0%

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

   6. Taylor expanded in x around -inf 97.0%

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

   if -1 < x < 1.5

   1. Initial program 23.4%

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

      1. +-commutative 23.4%

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

      2. hypot-1-def 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in x around 0 22.0%

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

      1. *-un-lft-identity 22.0%

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

      2. log-prod 22.0%

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

      3. metadata-eval 22.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} + \log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. +-commutative 22.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right) + 0}, x\right) \]

      5. log1p-def 98.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + 0.5 \cdot {x}^{2}\right)} + 0, x\right) \]

      6. +-commutative 98.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot {x}^{2} + \left|x\right|}\right) + 0, x\right) \]

      7. fma-def 98.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)}\right) + 0, x\right) \]

      8. add-sqr-sqrt 48.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right) + 0, x\right) \]

      9. fabs-sqr 48.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right) + 0, x\right) \]

      10. add-sqr-sqrt 98.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right)\right) + 0, x\right) \]

   6. Applied egg-rr 98.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 98.0%

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

      2. fma-udef 98.0%

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

      3. *-commutative 98.0%

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

      4. fma-udef 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in x around 0 98.3%

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

   if 1.5 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 6.4%

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

      1. *-un-lft-identity 6.4%

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

      2. log-prod 6.4%

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

      3. metadata-eval 6.4%

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

      4. +-commutative 6.4%

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

      5. add-sqr-sqrt 10.3%

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

      6. fabs-sqr 10.3%

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

      7. add-sqr-sqrt 6.4%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 96.0%

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

      11. mul-1-neg 96.0%

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

      12. neg-fabs 96.0%

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

      13. add-sqr-sqrt 96.0%

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

      14. fabs-sqr 96.0%

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

      15. add-sqr-sqrt 96.0%

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

   6. Applied egg-rr 96.0%

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

      1. +-rgt-identity 96.0%

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

      2. count-2 96.0%

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

      3. *-commutative 96.0%

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

   8. Simplified 96.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 5: 83.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- x)) x)
   (if (<= x 1.5) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 1.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(1.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(1.5))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 47.5%

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

      1. +-commutative 47.5%

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

      2. hypot-1-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in x around -inf 44.8%

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

   if -2 < x < 1.5

   1. Initial program 24.0%

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

      1. +-commutative 24.0%

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

      2. hypot-1-def 24.1%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in x around 0 22.1%

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

      1. *-un-lft-identity 22.1%

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

      2. log-prod 22.1%

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

      3. metadata-eval 22.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} + \log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. +-commutative 22.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right) + 0}, x\right) \]

      5. log1p-def 97.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + 0.5 \cdot {x}^{2}\right)} + 0, x\right) \]

      6. +-commutative 97.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot {x}^{2} + \left|x\right|}\right) + 0, x\right) \]

      7. fma-def 97.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)}\right) + 0, x\right) \]

      8. add-sqr-sqrt 47.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right) + 0, x\right) \]

      9. fabs-sqr 47.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right) + 0, x\right) \]

      10. add-sqr-sqrt 97.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right)\right) + 0, x\right) \]

   6. Applied egg-rr 97.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 97.4%

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

      2. fma-udef 97.4%

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

      3. *-commutative 97.4%

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

      4. fma-udef 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in x around 0 96.8%

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

   if 1.5 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 6.4%

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

      1. *-un-lft-identity 6.4%

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

      2. log-prod 6.4%

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

      3. metadata-eval 6.4%

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

      4. +-commutative 6.4%

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

      5. add-sqr-sqrt 10.3%

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

      6. fabs-sqr 10.3%

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

      7. add-sqr-sqrt 6.4%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 96.0%

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

      11. mul-1-neg 96.0%

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

      12. neg-fabs 96.0%

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

      13. add-sqr-sqrt 96.0%

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

      14. fabs-sqr 96.0%

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

      15. add-sqr-sqrt 96.0%

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

   6. Applied egg-rr 96.0%

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

      1. +-rgt-identity 96.0%

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

      2. count-2 96.0%

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

      3. *-commutative 96.0%

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

   8. Simplified 96.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 6: 83.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 1.5) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf((-1.0f / x)), x);
	} else if (x <= 1.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(Float32(-1.0) / x))), x);
	elseif (x <= Float32(1.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log((single(-1.0) / x)));
	elseif (x <= single(1.5))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 47.5%

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

      1. +-commutative 47.5%

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

      2. hypot-1-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in x around -inf 96.5%

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

   5. Taylor expanded in x around -inf 44.8%

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

      1. mul-1-neg 44.8%

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

   7. Simplified 44.8%

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

   if -2 < x < 1.5

   1. Initial program 24.0%

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

      1. +-commutative 24.0%

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

      2. hypot-1-def 24.1%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in x around 0 22.1%

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

      1. *-un-lft-identity 22.1%

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

      2. log-prod 22.1%

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

      3. metadata-eval 22.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} + \log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. +-commutative 22.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right) + 0}, x\right) \]

      5. log1p-def 97.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + 0.5 \cdot {x}^{2}\right)} + 0, x\right) \]

      6. +-commutative 97.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot {x}^{2} + \left|x\right|}\right) + 0, x\right) \]

      7. fma-def 97.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)}\right) + 0, x\right) \]

      8. add-sqr-sqrt 47.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right) + 0, x\right) \]

      9. fabs-sqr 47.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right) + 0, x\right) \]

      10. add-sqr-sqrt 97.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right)\right) + 0, x\right) \]

   6. Applied egg-rr 97.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 97.4%

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

      2. fma-udef 97.4%

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

      3. *-commutative 97.4%

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

      4. fma-udef 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in x around 0 96.8%

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

   if 1.5 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 6.4%

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

      1. *-un-lft-identity 6.4%

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

      2. log-prod 6.4%

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

      3. metadata-eval 6.4%

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

      4. +-commutative 6.4%

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

      5. add-sqr-sqrt 10.3%

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

      6. fabs-sqr 10.3%

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

      7. add-sqr-sqrt 6.4%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 96.0%

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

      11. mul-1-neg 96.0%

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

      12. neg-fabs 96.0%

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

      13. add-sqr-sqrt 96.0%

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

      14. fabs-sqr 96.0%

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

      15. add-sqr-sqrt 96.0%

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

   6. Applied egg-rr 96.0%

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

      1. +-rgt-identity 96.0%

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

      2. count-2 96.0%

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

      3. *-commutative 96.0%

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

   8. Simplified 96.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 7: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.5) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.5))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 48.3%

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

      1. +-commutative 48.3%

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

      2. hypot-1-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-lft-identity 98.6%

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

      2. +-commutative 98.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. fabs-sqr -0.0%

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

      5. add-sqr-sqrt 13.0%

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

   5. Applied egg-rr 13.0%

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

   6. Taylor expanded in x around -inf 97.0%

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

   if -1 < x < 1.5

   1. Initial program 23.4%

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

      1. +-commutative 23.4%

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

      2. hypot-1-def 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in x around 0 22.0%

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

      1. *-un-lft-identity 22.0%

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

      2. log-prod 22.0%

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

      3. metadata-eval 22.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} + \log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. +-commutative 22.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right) + 0}, x\right) \]

      5. log1p-def 98.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + 0.5 \cdot {x}^{2}\right)} + 0, x\right) \]

      6. +-commutative 98.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot {x}^{2} + \left|x\right|}\right) + 0, x\right) \]

      7. fma-def 98.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)}\right) + 0, x\right) \]

      8. add-sqr-sqrt 48.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right) + 0, x\right) \]

      9. fabs-sqr 48.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right) + 0, x\right) \]

      10. add-sqr-sqrt 98.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right)\right) + 0, x\right) \]

   6. Applied egg-rr 98.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 98.0%

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

      2. fma-udef 98.0%

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

      3. *-commutative 98.0%

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

      4. fma-udef 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in x around 0 97.3%

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

   if 1.5 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around -inf 6.4%

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

      1. *-un-lft-identity 6.4%

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

      2. log-prod 6.4%

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

      3. metadata-eval 6.4%

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

      4. +-commutative 6.4%

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

      5. add-sqr-sqrt 10.3%

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

      6. fabs-sqr 10.3%

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

      7. add-sqr-sqrt 6.4%

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

      8. add-sqr-sqrt -0.0%

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

      9. fabs-sqr -0.0%

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

      10. add-sqr-sqrt 96.0%

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

      11. mul-1-neg 96.0%

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

      12. neg-fabs 96.0%

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

      13. add-sqr-sqrt 96.0%

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

      14. fabs-sqr 96.0%

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

      15. add-sqr-sqrt 96.0%

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

   6. Applied egg-rr 96.0%

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

      1. +-rgt-identity 96.0%

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

      2. count-2 96.0%

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

      3. *-commutative 96.0%

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

   8. Simplified 96.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 8: 68.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.5) (copysign (log (- x)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -0.5f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.5))
		tmp = copysign(log(Float32(-x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -0.5

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

      2. hypot-1-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in x around -inf 44.1%

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

   if -0.5 < x

   1. Initial program 31.1%

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

      1. +-commutative 31.1%

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

      2. hypot-1-def 47.3%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in x around 0 27.6%

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

   6. Simplified 78.1%

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

      1. log1p-def 27.6%

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

      2. *-un-lft-identity 27.6%

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

      3. log-prod 27.6%

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

      4. metadata-eval 27.6%

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

      5. +-commutative 27.6%

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

      6. log1p-def 78.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0, x\right) \]

      7. add-sqr-sqrt 45.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + 0, x\right) \]

      8. fabs-sqr 45.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + 0, x\right) \]

      9. add-sqr-sqrt 78.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right) + 0, x\right) \]

   8. Applied egg-rr 78.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right) + 0}, x\right) \]
   9. Step-by-step derivation

      1. +-rgt-identity 78.1%

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

   10. Simplified 78.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 9: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 5.0) (copysign x x) (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 5.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(5.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(5.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5

   1. Initial program 32.9%

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

      1. +-commutative 32.9%

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

      2. hypot-1-def 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in x around 0 20.4%

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

      1. *-un-lft-identity 20.4%

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

      2. log-prod 20.4%

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

      3. metadata-eval 20.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} + \log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. +-commutative 20.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right) + 0}, x\right) \]

      5. log1p-def 69.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + 0.5 \cdot {x}^{2}\right)} + 0, x\right) \]

      6. +-commutative 69.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot {x}^{2} + \left|x\right|}\right) + 0, x\right) \]

      7. fma-def 69.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)}\right) + 0, x\right) \]

      8. add-sqr-sqrt 31.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right) + 0, x\right) \]

      9. fabs-sqr 31.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right) + 0, x\right) \]

      10. add-sqr-sqrt 69.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right)\right) + 0, x\right) \]

   6. Applied egg-rr 69.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 69.6%

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

      2. fma-udef 69.6%

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

      3. *-commutative 69.6%

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

      4. fma-udef 69.6%

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

   8. Simplified 69.6%

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

   9. Taylor expanded in x around 0 67.2%

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

   if 5 < x

   1. Initial program 47.5%

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

      1. +-commutative 47.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 44.7%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]

Alternative 10: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 1.5) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 1.5f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.5))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 31.9%

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

      1. +-commutative 31.9%

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

      2. hypot-1-def 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in x around 0 20.2%

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

      1. *-un-lft-identity 20.2%

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

      2. log-prod 20.2%

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

      3. metadata-eval 20.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0} + \log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. +-commutative 20.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right) + 0}, x\right) \]

      5. log1p-def 70.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + 0.5 \cdot {x}^{2}\right)} + 0, x\right) \]

      6. +-commutative 70.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot {x}^{2} + \left|x\right|}\right) + 0, x\right) \]

      7. fma-def 70.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)}\right) + 0, x\right) \]

      8. add-sqr-sqrt 31.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right) + 0, x\right) \]

      9. fabs-sqr 31.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right) + 0, x\right) \]

      10. add-sqr-sqrt 70.1%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right)\right) + 0, x\right) \]

   6. Applied egg-rr 70.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 70.1%

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

      2. fma-udef 70.1%

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

      3. *-commutative 70.1%

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

      4. fma-udef 70.1%

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

   8. Simplified 70.1%

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

   9. Taylor expanded in x around 0 67.8%

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

   if 1.5 < x

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 44.1%

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

   6. Simplified 44.1%

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

      1. log1p-def 44.1%

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

      2. *-un-lft-identity 44.1%

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

      3. log-prod 44.1%

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

      4. metadata-eval 44.1%

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

      5. +-commutative 44.1%

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

      6. log1p-def 44.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0, x\right) \]

      7. add-sqr-sqrt 44.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + 0, x\right) \]

      8. fabs-sqr 44.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + 0, x\right) \]

      9. add-sqr-sqrt 44.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right) + 0, x\right) \]

   8. Applied egg-rr 44.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right) + 0}, x\right) \]
   9. Step-by-step derivation

      1. +-rgt-identity 44.1%

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

   10. Simplified 44.1%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 11: 53.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 36.1%

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

   1. +-commutative 36.1%

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

   2. hypot-1-def 61.1%

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

3. Simplified 61.1%

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

4. Taylor expanded in x around 0 19.5%

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

   1. *-un-lft-identity 19.5%

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

   2. log-prod 19.5%

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

   3. metadata-eval 19.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0} + \log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

   4. +-commutative 19.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left(\left|x\right| + 0.5 \cdot {x}^{2}\right)\right) + 0}, x\right) \]

   5. log1p-def 57.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + 0.5 \cdot {x}^{2}\right)} + 0, x\right) \]

   6. +-commutative 57.8%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{0.5 \cdot {x}^{2} + \left|x\right|}\right) + 0, x\right) \]

   7. fma-def 57.8%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)}\right) + 0, x\right) \]

   8. add-sqr-sqrt 28.3%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right) + 0, x\right) \]

   9. fabs-sqr 28.3%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right) + 0, x\right) \]

   10. add-sqr-sqrt 57.7%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right)\right) + 0, x\right) \]

6. Applied egg-rr 57.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, {x}^{2}, x\right)\right) + 0}, x\right) \]
7. Step-by-step derivation

   1. +-rgt-identity 57.7%

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

   2. fma-udef 57.7%

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

   3. *-commutative 57.7%

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

   4. fma-udef 57.7%

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

8. Simplified 57.7%

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

9. Taylor expanded in x around 0 54.6%

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

10. Final simplification 54.6%

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

Developer target: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2023326 
(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))