Rust f32::asinh

Percentage Accurate: 38.2% → 99.5%

Time: 8.1s

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: 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.11599999666213989:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.11599999666213989)
     (copysign (log (/ -1.0 (- x (hypot 1.0 x)))) x)
     (if (<= t_0 0.10000000149011612)
       (copysign
        (* x (+ 1.0 (* (pow x 2.0) (- (* (* x x) 0.075) 0.16666666666666666))))
        x)
       (copysign (log1p (+ (hypot 1.0 x) (+ x -1.0))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.11599999666213989f) {
		tmp = copysignf(logf((-1.0f / (x - hypotf(1.0f, x)))), x);
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * (((x * x) * 0.075f) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(log1pf((hypotf(1.0f, x) + (x + -1.0f))), 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(-0.11599999666213989))
		tmp = copysign(log(Float32(Float32(-1.0) / Float32(x - hypot(Float32(1.0), x)))), x);
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32(Float32(x * x) * Float32(0.075)) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(log1p(Float32(hypot(Float32(1.0), x) + Float32(x + Float32(-1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -0.115999997

   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 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 9.1%

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

      2. div-sub 9.0%

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

      3. pow2 9.0%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 9.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 3.2%

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

   6. Applied egg-rr 11.6%

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

      1. div-sub 13.1%

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

      2. fma-undefine 13.1%

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

      3. unpow2 13.1%

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

      4. associate--r+ 47.6%

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

      5. sub-neg 47.6%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   if -0.115999997 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.100000001

   1. Initial program 22.2%

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

      1. +-commutative 22.2%

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

      2. hypot-1-def 22.3%

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

   3. Simplified 22.3%

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

      1. log1p-expm1-u 22.3%

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

      2. expm1-undefine 22.3%

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

      3. add-exp-log 22.3%

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

      4. add-sqr-sqrt 10.5%

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

      5. fabs-sqr 10.5%

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

      6. add-sqr-sqrt 22.6%

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

   6. Applied egg-rr 22.6%

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

   7. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   9. Applied egg-rr 99.9%

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

   if 0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 49.2%

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

      1. +-commutative 49.2%

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

      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. Add Preprocessing
   5. Step-by-step derivation

      1. log1p-expm1-u 99.9%

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

      2. expm1-undefine 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 100.0%

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

      5. fabs-sqr 100.0%

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

      6. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. associate--l+ 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.11599999666213989:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x + -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.11800000071525574:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.11800000071525574)
   (copysign (log (/ -1.0 (- x (hypot 1.0 x)))) x)
   (if (<= x 0.10000000149011612)
     (copysign
      (* x (+ 1.0 (* (pow x 2.0) (- (* (* x x) 0.075) 0.16666666666666666))))
      x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.11800000071525574f) {
		tmp = copysignf(logf((-1.0f / (x - hypotf(1.0f, x)))), x);
	} else if (x <= 0.10000000149011612f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * (((x * x) * 0.075f) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.11800000071525574))
		tmp = copysign(log(Float32(Float32(-1.0) / Float32(x - hypot(Float32(1.0), x)))), x);
	elseif (x <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32(Float32(x * x) * Float32(0.075)) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.11800000071525574))
		tmp = sign(x) * abs(log((single(-1.0) / (x - hypot(single(1.0), x)))));
	elseif (x <= single(0.10000000149011612))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x * x) * single(0.075)) - single(0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.118000001

   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 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 9.1%

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

      2. div-sub 9.0%

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

      3. pow2 9.0%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 9.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. add-sqr-sqrt 3.2%

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

   6. Applied egg-rr 11.6%

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

      1. div-sub 13.1%

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

      2. fma-undefine 13.1%

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

      3. unpow2 13.1%

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

      4. associate--r+ 47.6%

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

      5. sub-neg 47.6%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   if -0.118000001 < x < 0.100000001

   1. Initial program 22.2%

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

      1. +-commutative 22.2%

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

      2. hypot-1-def 22.3%

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

   3. Simplified 22.3%

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

      1. log1p-expm1-u 22.3%

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

      2. expm1-undefine 22.3%

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

      3. add-exp-log 22.3%

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

      4. add-sqr-sqrt 10.5%

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

      5. fabs-sqr 10.5%

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

      6. add-sqr-sqrt 22.6%

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

   6. Applied egg-rr 22.6%

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

   7. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   9. Applied egg-rr 99.9%

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

   if 0.100000001 < x

   1. Initial program 49.2%

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

      1. +-commutative 49.2%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 49.2%

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

      1. rem-square-sqrt 49.2%

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

      2. fabs-sqr 49.2%

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

      3. rem-square-sqrt 49.2%

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

      4. metadata-eval 49.2%

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

      5. unpow2 49.2%

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

      6. hypot-undefine 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.11800000071525574:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.20000000298023224)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.10000000149011612)
     (copysign
      (* x (+ 1.0 (* (pow x 2.0) (- (* (* x x) 0.075) 0.16666666666666666))))
      x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.20000000298023224f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.10000000149011612f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * (((x * x) * 0.075f) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.20000000298023224))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32(Float32(x * x) * Float32(0.075)) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.20000000298023224))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.10000000149011612))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x * x) * single(0.075)) - single(0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.200000003

   1. Initial program 50.3%

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

      1. +-commutative 50.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 7.8%

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

      2. frac-2neg 7.8%

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

      3. log-div 7.8%

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

   6. Applied egg-rr 11.7%

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

      1. fma-undefine 11.7%

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

      2. unpow2 11.7%

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

      3. associate--r+ 46.8%

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

      4. sub-neg 46.8%

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

      5. +-inverses 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. neg-sub0 100.0%

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

      14. +-commutative 100.0%

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

      15. sub-neg 100.0%

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

   8. Simplified 100.0%

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

   if -0.200000003 < x < 0.100000001

   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 22.8%

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

   3. Simplified 22.8%

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

      1. log1p-expm1-u 22.8%

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

      2. expm1-undefine 22.8%

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

      3. add-exp-log 22.8%

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

      4. add-sqr-sqrt 10.4%

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

      5. fabs-sqr 10.4%

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

      6. add-sqr-sqrt 23.2%

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

   6. Applied egg-rr 23.2%

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

   7. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   9. Applied egg-rr 99.9%

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

   if 0.100000001 < x

   1. Initial program 49.2%

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

      1. +-commutative 49.2%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 49.2%

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

      1. rem-square-sqrt 49.2%

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

      2. fabs-sqr 49.2%

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

      3. rem-square-sqrt 49.2%

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

      4. metadata-eval 49.2%

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

      5. unpow2 49.2%

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

      6. hypot-undefine 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.10000000149011612))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x * x) * single(0.075)) - single(0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 10.9%

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

      2. *-un-lft-identity 10.9%

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

      3. exp-prod 10.9%

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

      4. log-pow 99.8%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 12.2%

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

   6. Applied egg-rr 12.2%

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

      1. rem-log-exp 12.2%

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

      2. *-rgt-identity 12.2%

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

      3. +-commutative 12.2%

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

   8. Applied egg-rr 12.2%

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

   9. Taylor expanded in x around -inf 97.4%

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

   if -5 < x < 0.100000001

   1. Initial program 23.3%

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

      1. +-commutative 23.3%

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

      2. hypot-1-def 23.4%

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

   3. Simplified 23.4%

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

      1. log1p-expm1-u 23.4%

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

      2. expm1-undefine 23.4%

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

      3. add-exp-log 23.4%

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

      4. add-sqr-sqrt 10.4%

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

      5. fabs-sqr 10.4%

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

      6. add-sqr-sqrt 23.7%

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

   6. Applied egg-rr 23.7%

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

   7. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 0.100000001 < x

   1. Initial program 49.2%

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

      1. +-commutative 49.2%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 49.2%

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

      1. rem-square-sqrt 49.2%

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

      2. fabs-sqr 49.2%

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

      3. rem-square-sqrt 49.2%

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

      4. metadata-eval 49.2%

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

      5. unpow2 49.2%

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

      6. hypot-undefine 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.1%

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

Alternative 5: 98.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32(Float32(x * x) * Float32(0.075)) - 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(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x * x) * single(0.075)) - 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 < -5

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 10.9%

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

      2. *-un-lft-identity 10.9%

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

      3. exp-prod 10.9%

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

      4. log-pow 99.8%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 12.2%

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

   6. Applied egg-rr 12.2%

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

      1. rem-log-exp 12.2%

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

      2. *-rgt-identity 12.2%

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

      3. +-commutative 12.2%

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

   8. Applied egg-rr 12.2%

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

   9. Taylor expanded in x around -inf 97.4%

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

   if -5 < x < 2

   1. Initial program 24.5%

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

      1. +-commutative 24.5%

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

      2. hypot-1-def 24.5%

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

   3. Simplified 24.5%

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

      1. log1p-expm1-u 24.5%

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

      2. expm1-undefine 24.5%

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

      3. add-exp-log 24.5%

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

      4. add-sqr-sqrt 11.7%

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

      5. fabs-sqr 11.7%

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

      6. add-sqr-sqrt 24.8%

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

   6. Applied egg-rr 24.8%

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

   7. Taylor expanded in x around 0 98.7%

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

      1. unpow2 98.7%

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

   9. Applied egg-rr 98.7%

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

   if 2 < x

   1. Initial program 47.4%

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

      1. +-commutative 47.4%

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

      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. Add Preprocessing

   5. Taylor expanded in x around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. rem-square-sqrt 98.9%

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

      3. fabs-sqr 98.9%

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

      4. rem-square-sqrt 98.9%

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

      5. *-inverses 98.9%

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

      6. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 98.4%

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

Alternative 6: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.2000000476837158:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.2000000476837158)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.2000000476837158f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.2000000476837158))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.2000000476837158))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 10.9%

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

      2. *-un-lft-identity 10.9%

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

      3. exp-prod 10.9%

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

      4. log-pow 99.8%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 12.2%

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

   6. Applied egg-rr 12.2%

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

      1. rem-log-exp 12.2%

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

      2. *-rgt-identity 12.2%

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

      3. +-commutative 12.2%

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

   8. Applied egg-rr 12.2%

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

   9. Taylor expanded in x around -inf 97.4%

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

   if -5 < x < 1.20000005

   1. Initial program 23.9%

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

      1. +-commutative 23.9%

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

      2. hypot-1-def 23.9%

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

   3. Simplified 23.9%

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

      1. log1p-expm1-u 24.0%

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

      2. expm1-undefine 24.0%

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

      3. add-exp-log 24.0%

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

      4. add-sqr-sqrt 11.0%

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

      5. fabs-sqr 11.0%

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

      6. add-sqr-sqrt 24.3%

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

   6. Applied egg-rr 24.3%

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

   7. Taylor expanded in x around 0 98.4%

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

      1. distribute-rgt-in 98.4%

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

      2. *-lft-identity 98.4%

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

      3. associate-*l* 98.4%

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

      4. unpow2 98.4%

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

      5. unpow3 98.4%

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

   9. Simplified 98.4%

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

   if 1.20000005 < x

   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 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. Add Preprocessing

   5. Taylor expanded in x around inf 97.8%

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

      1. +-commutative 97.8%

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

      2. rem-square-sqrt 97.8%

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

      3. fabs-sqr 97.8%

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

      4. rem-square-sqrt 97.8%

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

      5. *-inverses 97.8%

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

      6. metadata-eval 97.8%

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

   7. Simplified 97.8%

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

Alternative 7: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(0.125 + x \cdot -0.05\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 2.0)
     (copysign
      (*
       x
       (+ 1.0 (* (* x x) (- (* x (+ 0.125 (* x -0.05))) 0.16666666666666666))))
      x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 2.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * ((x * (0.125f + (x * -0.05f))) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(x * Float32(Float32(0.125) + Float32(x * Float32(-0.05)))) - 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(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * ((x * (single(0.125) + (x * single(-0.05)))) - 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 < -5

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 10.9%

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

      2. *-un-lft-identity 10.9%

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

      3. exp-prod 10.9%

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

      4. log-pow 99.8%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 12.2%

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

   6. Applied egg-rr 12.2%

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

      1. rem-log-exp 12.2%

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

      2. *-rgt-identity 12.2%

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

      3. +-commutative 12.2%

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

   8. Applied egg-rr 12.2%

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

   9. Taylor expanded in x around -inf 97.4%

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

   if -5 < x < 2

   1. Initial program 24.5%

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

      1. +-commutative 24.5%

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

      2. hypot-1-def 24.5%

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

   3. Simplified 24.5%

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

      1. log1p-expm1-u 24.5%

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

      2. expm1-undefine 24.5%

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

      3. add-exp-log 24.5%

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

      4. add-sqr-sqrt 11.7%

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

      5. fabs-sqr 11.7%

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

      6. add-sqr-sqrt 24.8%

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

   6. Applied egg-rr 24.8%

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

   7. Taylor expanded in x around 0 97.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)}\right), x\right) \]
   8. Step-by-step derivation

      1. *-commutative 97.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)\right), x\right) \]

   9. Simplified 97.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + x \cdot 0.5\right)}\right), x\right) \]

   10. Taylor expanded in x around 0 97.1%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot \left(0.125 + -0.05 \cdot x\right) - 0.16666666666666666\right)\right)}, x\right) \]
   11. Step-by-step derivation

       1. unpow2 98.7%

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

   12. Applied egg-rr 97.1%

       \[\leadsto \mathsf{copysign}\left(x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(0.125 + -0.05 \cdot x\right) - 0.16666666666666666\right)\right), x\right) \]

   if 2 < x

   1. Initial program 47.4%

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

      1. +-commutative 47.4%

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

      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. Add Preprocessing

   5. Taylor expanded in x around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. rem-square-sqrt 98.9%

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

      3. fabs-sqr 98.9%

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

      4. rem-square-sqrt 98.9%

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

      5. *-inverses 98.9%

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

      6. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(0.125 + x \cdot -0.05\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(0.125 + x \cdot -0.05\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (- x)) x)
   (if (<= x 2.0)
     (copysign
      (*
       x
       (+ 1.0 (* (* x x) (- (* x (+ 0.125 (* x -0.05))) 0.16666666666666666))))
      x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 2.0f) {
		tmp = copysignf((x * (1.0f + ((x * x) * ((x * (0.125f + (x * -0.05f))) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(x * Float32(Float32(0.125) + Float32(x * Float32(-0.05)))) - 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(-5.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs((x * (single(1.0) + ((x * x) * ((x * (single(0.125) + (x * single(-0.05)))) - 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 < -5

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around -inf 44.4%

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

      1. mul-1-neg 44.4%

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

   7. Simplified 44.4%

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

   if -5 < x < 2

   1. Initial program 24.5%

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

      1. +-commutative 24.5%

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

      2. hypot-1-def 24.5%

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

   3. Simplified 24.5%

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

      1. log1p-expm1-u 24.5%

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

      2. expm1-undefine 24.5%

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

      3. add-exp-log 24.5%

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

      4. add-sqr-sqrt 11.7%

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

      5. fabs-sqr 11.7%

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

      6. add-sqr-sqrt 24.8%

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

   6. Applied egg-rr 24.8%

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

   7. Taylor expanded in x around 0 97.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)}\right), x\right) \]
   8. Step-by-step derivation

      1. *-commutative 97.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)\right), x\right) \]

   9. Simplified 97.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + x \cdot 0.5\right)}\right), x\right) \]

   10. Taylor expanded in x around 0 97.1%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot \left(0.125 + -0.05 \cdot x\right) - 0.16666666666666666\right)\right)}, x\right) \]
   11. Step-by-step derivation

       1. unpow2 98.7%

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

   12. Applied egg-rr 97.1%

       \[\leadsto \mathsf{copysign}\left(x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(0.125 + -0.05 \cdot x\right) - 0.16666666666666666\right)\right), x\right) \]

   if 2 < x

   1. Initial program 47.4%

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

      1. +-commutative 47.4%

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

      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. Add Preprocessing

   5. Taylor expanded in x around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. rem-square-sqrt 98.9%

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

      3. fabs-sqr 98.9%

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

      4. rem-square-sqrt 98.9%

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

      5. *-inverses 98.9%

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

      6. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(0.125 + x \cdot -0.05\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0) (copysign (log (- x)) x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		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 < -5

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around -inf 44.4%

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

      1. mul-1-neg 44.4%

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

   7. Simplified 44.4%

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

   if -5 < x

   1. Initial program 31.2%

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

      1. +-commutative 31.2%

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

      2. hypot-1-def 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in x around 0 26.9%

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

      1. log1p-define 78.0%

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

      2. rem-square-sqrt 45.4%

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

      3. fabs-sqr 45.4%

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

      4. rem-square-sqrt 77.9%

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

   7. Simplified 77.9%

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

Alternative 10: 62.5% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 2.0) (copysign x x) (copysign (log1p x) x)))

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. hypot-1-def 48.4%

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

   3. Simplified 48.4%

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

      1. log1p-expm1-u 48.4%

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

      2. expm1-undefine 48.4%

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

      3. add-exp-log 48.4%

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

      4. add-sqr-sqrt 8.0%

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

      5. fabs-sqr 8.0%

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

      6. add-sqr-sqrt 20.8%

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

   6. Applied egg-rr 20.8%

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

   7. Taylor expanded in x around 0 69.1%

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

   if 2 < x

   1. Initial program 47.4%

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

      1. +-commutative 47.4%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 44.6%

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

      1. log1p-define 44.6%

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

      2. rem-square-sqrt 44.6%

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

      3. fabs-sqr 44.6%

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

      4. rem-square-sqrt 44.6%

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

   7. Simplified 44.6%

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

Alternative 11: 54.3% 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 35.7%

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

   1. +-commutative 35.7%

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

   2. hypot-1-def 59.9%

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

3. Simplified 59.9%

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

   1. log1p-expm1-u 59.9%

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

   2. expm1-undefine 59.9%

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

   3. add-exp-log 59.9%

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

   4. add-sqr-sqrt 28.5%

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

   5. fabs-sqr 28.5%

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

   6. add-sqr-sqrt 38.4%

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

6. Applied egg-rr 38.4%

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

7. Taylor expanded in x around 0 56.1%

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

Developer Target 1: 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 2024113 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (! :herbie-platform default (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

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