Rust f32::asinh

Percentage Accurate: 37.4% → 98.6%

Time: 9.9s

Alternatives: 11

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := {\left(x + 1\right)}^{2}\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(0.001388888888888889, {x}^{2} \cdot \left(\frac{45}{x + 1} + \left(\frac{45}{t\_1} + \frac{30}{{\left(x + 1\right)}^{3}}\right)\right), \frac{-0.125}{x + 1} + \frac{-0.125}{t\_1}\right), \frac{0.5}{x + 1}\right), \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \log 2, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (pow (+ x 1.0) 2.0)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (log (/ -1.0 (- x (hypot 1.0 x)))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (fma
         (pow x 2.0)
         (fma
          (pow x 2.0)
          (fma
           0.001388888888888889
           (*
            (pow x 2.0)
            (+
             (/ 45.0 (+ x 1.0))
             (+ (/ 45.0 t_1) (/ 30.0 (pow (+ x 1.0) 3.0)))))
           (+ (/ -0.125 (+ x 1.0)) (/ -0.125 t_1)))
          (/ 0.5 (+ x 1.0)))
         (log1p x))
        x)
       (copysign (+ (log x) (log 2.0)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = powf((x + 1.0f), 2.0f);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = copysignf(logf((-1.0f / (x - hypotf(1.0f, x)))), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(fmaf(powf(x, 2.0f), fmaf(powf(x, 2.0f), fmaf(0.001388888888888889f, (powf(x, 2.0f) * ((45.0f / (x + 1.0f)) + ((45.0f / t_1) + (30.0f / powf((x + 1.0f), 3.0f))))), ((-0.125f / (x + 1.0f)) + (-0.125f / t_1))), (0.5f / (x + 1.0f))), log1pf(x)), x);
	} else {
		tmp = copysignf((logf(x) + logf(2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = Float32(x + Float32(1.0)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = copysign(log(Float32(Float32(-1.0) / Float32(x - hypot(Float32(1.0), x)))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(fma((x ^ Float32(2.0)), fma((x ^ Float32(2.0)), fma(Float32(0.001388888888888889), Float32((x ^ Float32(2.0)) * Float32(Float32(Float32(45.0) / Float32(x + Float32(1.0))) + Float32(Float32(Float32(45.0) / t_1) + Float32(Float32(30.0) / (Float32(x + Float32(1.0)) ^ Float32(3.0)))))), Float32(Float32(Float32(-0.125) / Float32(x + Float32(1.0))) + Float32(Float32(-0.125) / t_1))), Float32(Float32(0.5) / Float32(x + Float32(1.0)))), log1p(x)), x);
	else
		tmp = copysign(Float32(log(x) + log(Float32(2.0))), 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) < -0.200000003

   1. Initial program 57.4%

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

      1. flip-+ 15.0%

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

      2. add-sqr-sqrt 14.9%

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

      3. div-sub 15.1%

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

      4. pow2 15.1%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 15.1%

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

      8. +-commutative 15.1%

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

      9. hypot-1-def 15.1%

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

      10. add-sqr-sqrt -0.0%

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

      11. fabs-sqr -0.0%

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

      12. add-sqr-sqrt 5.5%

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

      13. fma-define 5.5%

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

   4. Applied egg-rr 17.4%

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

      1. div-sub 20.9%

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

      2. remove-double-neg 20.9%

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

      3. remove-double-neg 20.9%

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

      4. fma-undefine 20.8%

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

      5. unpow2 20.8%

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

      6. associate--r+ 54.6%

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

      7. +-inverses 99.8%

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

      8. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 22.9%

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

   5. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(0.001388888888888889, {x}^{2} \cdot \left(\frac{45}{1 + x} + \left(\frac{45}{{\left(1 + x\right)}^{2}} + \frac{30}{{\left(1 + x\right)}^{3}}\right)\right), \frac{-0.125}{1 + x} + \frac{-0.125}{{\left(1 + x\right)}^{2}}\right), \frac{0.5}{1 + x}\right), \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. rem-square-sqrt 98.3%

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

      2. fabs-sqr 98.3%

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

      3. rem-square-sqrt 98.3%

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

   5. Simplified 98.3%

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

      1. log-prod 98.9%

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

      2. *-inverses 98.9%

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

      3. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(0.001388888888888889, {x}^{2} \cdot \left(\frac{45}{x + 1} + \left(\frac{45}{{\left(x + 1\right)}^{2}} + \frac{30}{{\left(x + 1\right)}^{3}}\right)\right), \frac{-0.125}{x + 1} + \frac{-0.125}{{\left(x + 1\right)}^{2}}\right), \frac{0.5}{x + 1}\right), \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \log 2, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \log 2, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.5)
     (copysign (log (/ -1.0 (- x (hypot 1.0 x)))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (pow x 2.0)
           (-
            (* (pow x 2.0) (+ 0.075 (* (pow x 2.0) -0.044642857142857144)))
            0.16666666666666666))))
        x)
       (copysign (+ (log x) (log 2.0)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.5f) {
		tmp = copysignf(logf((-1.0f / (x - hypotf(1.0f, x)))), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((powf(x, 2.0f) * (0.075f + (powf(x, 2.0f) * -0.044642857142857144f))) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf((logf(x) + logf(2.0f)), 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(-0.5))
		tmp = copysign(log(Float32(Float32(-1.0) / Float32(x - hypot(Float32(1.0), x)))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(Float32(0.075) + Float32((x ^ Float32(2.0)) * Float32(-0.044642857142857144)))) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(Float32(log(x) + log(Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.5))
		tmp = sign(x) * abs(log((single(-1.0) / (x - hypot(single(1.0), x)))));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x ^ single(2.0)) * (single(0.075) + ((x ^ single(2.0)) * single(-0.044642857142857144)))) - single(0.16666666666666666))))));
	else
		tmp = sign(x) * abs((log(x) + log(single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -0.5

   1. Initial program 55.9%

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

      1. flip-+ 11.9%

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

      2. add-sqr-sqrt 11.9%

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

      3. div-sub 11.9%

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

      4. pow2 11.9%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 11.9%

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

      8. +-commutative 11.9%

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

      9. hypot-1-def 11.9%

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

      10. add-sqr-sqrt -0.0%

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

      11. fabs-sqr -0.0%

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

      12. add-sqr-sqrt 4.3%

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

      13. fma-define 4.3%

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

   4. Applied egg-rr 14.4%

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

      1. div-sub 18.1%

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

      2. remove-double-neg 18.1%

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

      3. remove-double-neg 18.1%

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

      4. fma-undefine 18.0%

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

      5. unpow2 18.0%

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

      6. associate--r+ 52.9%

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

      7. +-inverses 100.0%

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

      8. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 23.0%

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

      1. flip-+ 23.0%

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

      2. add-sqr-sqrt 22.9%

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

      3. div-sub 22.8%

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

      4. pow2 22.8%

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

      5. add-sqr-sqrt 10.5%

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

      6. fabs-sqr 10.5%

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

      7. add-sqr-sqrt 22.8%

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

      8. +-commutative 22.8%

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

      9. hypot-1-def 22.8%

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

      10. add-sqr-sqrt 10.5%

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

      11. fabs-sqr 10.5%

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

      12. add-sqr-sqrt 21.0%

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

      13. fma-define 21.0%

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

   4. Applied egg-rr 22.6%

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

      1. div-sub 23.0%

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

      2. remove-double-neg 23.0%

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

      3. remove-double-neg 23.0%

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

      4. fma-undefine 23.0%

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

      5. unpow2 23.0%

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

      6. associate--r+ 23.0%

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

      7. +-inverses 23.0%

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

      8. metadata-eval 23.0%

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

   6. Simplified 23.0%

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

   7. Taylor expanded in x around 0 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right)}, x\right) \]

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. rem-square-sqrt 98.3%

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

      2. fabs-sqr 98.3%

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

      3. rem-square-sqrt 98.3%

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

   5. Simplified 98.3%

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

      1. log-prod 98.9%

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

      2. *-inverses 98.9%

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

      3. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \log 2, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \log 2, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.20000000298023224)
     (copysign (log (/ -1.0 (- x (hypot 1.0 x)))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (*
         x
         (+ 1.0 (* (pow x 2.0) (- (* (pow x 2.0) 0.075) 0.16666666666666666))))
        x)
       (copysign (+ (log x) (log 2.0)) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = copysignf(logf((-1.0f / (x - hypotf(1.0f, x)))), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((powf(x, 2.0f) * 0.075f) - 0.16666666666666666f)))), x);
	} else {
		tmp = copysignf((logf(x) + logf(2.0f)), 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(-0.20000000298023224))
		tmp = copysign(log(Float32(Float32(-1.0) / Float32(x - hypot(Float32(1.0), x)))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32((x ^ Float32(2.0)) * Float32(0.075)) - Float32(0.16666666666666666))))), x);
	else
		tmp = copysign(Float32(log(x) + log(Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.20000000298023224))
		tmp = sign(x) * abs(log((single(-1.0) / (x - hypot(single(1.0), x)))));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * (((x ^ single(2.0)) * single(0.075)) - single(0.16666666666666666))))));
	else
		tmp = sign(x) * abs((log(x) + log(single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -0.200000003

   1. Initial program 57.4%

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

      1. flip-+ 15.0%

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

      2. add-sqr-sqrt 14.9%

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

      3. div-sub 15.1%

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

      4. pow2 15.1%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 15.1%

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

      8. +-commutative 15.1%

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

      9. hypot-1-def 15.1%

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

      10. add-sqr-sqrt -0.0%

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

      11. fabs-sqr -0.0%

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

      12. add-sqr-sqrt 5.5%

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

      13. fma-define 5.5%

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

   4. Applied egg-rr 17.4%

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

      1. div-sub 20.9%

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

      2. remove-double-neg 20.9%

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

      3. remove-double-neg 20.9%

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

      4. fma-undefine 20.8%

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

      5. unpow2 20.8%

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

      6. associate--r+ 54.6%

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

      7. +-inverses 99.8%

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

      8. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   if -0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.200000003

   1. Initial program 21.8%

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

      1. flip-+ 21.9%

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

      2. add-sqr-sqrt 21.8%

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

      3. div-sub 21.7%

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

      4. pow2 21.7%

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

      5. add-sqr-sqrt 10.7%

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

      6. fabs-sqr 10.7%

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

      7. add-sqr-sqrt 21.7%

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

      8. +-commutative 21.7%

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

      9. hypot-1-def 21.7%

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

      10. add-sqr-sqrt 10.7%

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

      11. fabs-sqr 10.7%

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

      12. add-sqr-sqrt 20.7%

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

      13. fma-define 20.7%

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

   4. Applied egg-rr 21.6%

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

      1. div-sub 21.9%

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

      2. remove-double-neg 21.9%

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

      3. remove-double-neg 21.9%

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

      4. fma-undefine 21.9%

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

      5. unpow2 21.9%

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

      6. associate--r+ 21.9%

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

      7. +-inverses 21.9%

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

      8. metadata-eval 21.9%

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

   6. Simplified 21.9%

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

   7. Taylor expanded in x around 0 99.8%

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

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. rem-square-sqrt 98.3%

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

      2. fabs-sqr 98.3%

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

      3. rem-square-sqrt 98.3%

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

   5. Simplified 98.3%

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

      1. log-prod 98.9%

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

      2. *-inverses 98.9%

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

      3. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 99.6%

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

Alternative 4: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.0%

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

   3. Taylor expanded in x around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. *-commutative 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. mul-1-neg 98.1%

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

      5. unsub-neg 98.1%

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

      6. sub-neg 98.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 13.1%

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

      10. associate-*r/ 13.1%

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

      11. metadata-eval 13.1%

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

      12. distribute-neg-frac 13.1%

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

      13. metadata-eval 13.1%

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

   5. Simplified 13.1%

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

   6. Taylor expanded in x around 0 96.4%

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

   if -1 < x < 0.100000001

   1. Initial program 23.0%

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

   3. Taylor expanded in x around 0 21.6%

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

      1. +-commutative 21.6%

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

      2. fma-define 21.6%

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

      3. rem-square-sqrt 9.6%

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

      4. fabs-sqr 9.6%

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

      5. rem-square-sqrt 21.8%

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

   5. Simplified 21.8%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. distribute-rgt-in 98.8%

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

      2. *-lft-identity 98.8%

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

      3. associate-*l* 98.8%

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

      4. unpow2 98.8%

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

      5. unpow3 98.8%

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

   8. Simplified 98.8%

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

   if 0.100000001 < x

   1. Initial program 57.0%

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

      1. *-un-lft-identity 57.0%

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

      2. *-commutative 57.0%

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

      3. log-prod 57.0%

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

      4. *-un-lft-identity 57.0%

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

      5. *-un-lft-identity 57.0%

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

      6. add-sqr-sqrt 57.0%

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

      7. fabs-sqr 57.0%

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

      8. add-sqr-sqrt 57.0%

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

      9. +-commutative 57.0%

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

      10. hypot-1-def 98.6%

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

      11. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. +-rgt-identity 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 98.3%

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

Alternative 5: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.03999999910593033)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.10000000149011612)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.03999999910593033f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.10000000149011612f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.03999999910593033))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.10000000149011612))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.03999999910593033))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.10000000149011612))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0399999991

   1. Initial program 57.9%

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

      1. flip-+ 16.5%

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

      2. frac-2neg 16.5%

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

      3. log-div 16.3%

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

      4. pow2 16.3%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt 16.3%

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

      8. add-sqr-sqrt 16.4%

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

      9. fma-define 16.4%

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

   4. Applied egg-rr 22.3%

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

      1. neg-sub0 22.3%

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

      2. associate--r- 22.3%

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

      3. neg-sub0 22.3%

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

      4. +-commutative 22.3%

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

      5. fma-undefine 22.3%

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

      6. unpow2 22.3%

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

      7. +-commutative 22.3%

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

      8. associate-+l+ 55.3%

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

      9. sub-neg 55.3%

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

      10. +-inverses 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. neg-sub0 99.8%

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

      14. neg-sub0 99.8%

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

      15. associate--r- 99.8%

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

      16. neg-sub0 99.8%

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

      17. +-commutative 99.8%

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

      18. sub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -0.0399999991 < x < 0.100000001

   1. Initial program 20.8%

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

   3. Taylor expanded in x around 0 20.4%

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

      1. +-commutative 20.4%

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

      2. fma-define 20.4%

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

      3. rem-square-sqrt 9.9%

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

      4. fabs-sqr 9.9%

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

      5. rem-square-sqrt 20.7%

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

   5. Simplified 20.7%

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

   6. Taylor expanded in x around 0 99.7%

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

      1. distribute-rgt-in 99.8%

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

      2. *-lft-identity 99.8%

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

      3. associate-*l* 99.8%

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

      4. unpow2 99.8%

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

      5. unpow3 99.8%

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

   8. Simplified 99.8%

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

   if 0.100000001 < x

   1. Initial program 57.0%

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

      1. *-un-lft-identity 57.0%

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

      2. *-commutative 57.0%

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

      3. log-prod 57.0%

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

      4. *-un-lft-identity 57.0%

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

      5. *-un-lft-identity 57.0%

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

      6. add-sqr-sqrt 57.0%

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

      7. fabs-sqr 57.0%

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

      8. add-sqr-sqrt 57.0%

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

      9. +-commutative 57.0%

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

      10. hypot-1-def 98.6%

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

      11. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. +-rgt-identity 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \log 2, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.20000000298023224)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (+ (log x) (log 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf((logf(x) + logf(2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(Float32(log(x) + log(Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs((log(x) + log(single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.0%

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

   3. Taylor expanded in x around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. *-commutative 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. mul-1-neg 98.1%

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

      5. unsub-neg 98.1%

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

      6. sub-neg 98.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 13.1%

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

      10. associate-*r/ 13.1%

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

      11. metadata-eval 13.1%

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

      12. distribute-neg-frac 13.1%

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

      13. metadata-eval 13.1%

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

   5. Simplified 13.1%

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

   6. Taylor expanded in x around 0 96.4%

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

   if -1 < x < 0.200000003

   1. Initial program 23.5%

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

   3. Taylor expanded in x around 0 21.9%

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

      1. +-commutative 21.9%

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

      2. fma-define 21.9%

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

      3. rem-square-sqrt 10.0%

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

      4. fabs-sqr 10.0%

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

      5. rem-square-sqrt 22.1%

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

   5. Simplified 22.1%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. distribute-rgt-in 98.6%

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

      2. *-lft-identity 98.6%

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

      3. associate-*l* 98.6%

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

      4. unpow2 98.6%

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

      5. unpow3 98.6%

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

   8. Simplified 98.6%

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

   if 0.200000003 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. rem-square-sqrt 98.3%

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

      2. fabs-sqr 98.3%

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

      3. rem-square-sqrt 98.3%

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

   5. Simplified 98.3%

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

      1. log-prod 98.9%

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

      2. *-inverses 98.9%

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

      3. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x + \log 2, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.20000000298023224)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.20000000298023224f) {
		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(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.20000000298023224))
		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(-1.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.20000000298023224))
		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 < -1

   1. Initial program 55.0%

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

   3. Taylor expanded in x around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. *-commutative 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. mul-1-neg 98.1%

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

      5. unsub-neg 98.1%

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

      6. sub-neg 98.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 13.1%

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

      10. associate-*r/ 13.1%

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

      11. metadata-eval 13.1%

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

      12. distribute-neg-frac 13.1%

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

      13. metadata-eval 13.1%

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

   5. Simplified 13.1%

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

   6. Taylor expanded in x around 0 96.4%

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

   if -1 < x < 0.200000003

   1. Initial program 23.5%

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

   3. Taylor expanded in x around 0 21.9%

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

      1. +-commutative 21.9%

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

      2. fma-define 21.9%

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

      3. rem-square-sqrt 10.0%

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

      4. fabs-sqr 10.0%

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

      5. rem-square-sqrt 22.1%

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

   5. Simplified 22.1%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. distribute-rgt-in 98.6%

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

      2. *-lft-identity 98.6%

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

      3. associate-*l* 98.6%

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

      4. unpow2 98.6%

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

      5. unpow3 98.6%

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

   8. Simplified 98.6%

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

   if 0.200000003 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. rem-square-sqrt 98.3%

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

      2. fabs-sqr 98.3%

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

      3. rem-square-sqrt 98.3%

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

   5. Simplified 98.3%

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

      1. *-un-lft-identity 98.3%

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

      2. log-prod 98.3%

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

      3. metadata-eval 98.3%

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

      4. *-inverses 98.3%

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

      5. metadata-eval 98.3%

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

   7. Applied egg-rr 98.3%

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

      1. +-lft-identity 98.3%

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

   9. Simplified 98.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.20000000298023224)
     (copysign x x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-1.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.0%

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

   3. Taylor expanded in x around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. *-commutative 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. mul-1-neg 98.1%

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

      5. unsub-neg 98.1%

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

      6. sub-neg 98.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 13.1%

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

      10. associate-*r/ 13.1%

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

      11. metadata-eval 13.1%

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

      12. distribute-neg-frac 13.1%

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

      13. metadata-eval 13.1%

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

   5. Simplified 13.1%

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

   6. Taylor expanded in x around 0 96.4%

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

   if -1 < x < 0.200000003

   1. Initial program 23.5%

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

   3. Taylor expanded in x around 0 21.9%

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

      1. +-commutative 21.9%

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

      2. fma-define 21.9%

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

      3. rem-square-sqrt 10.0%

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

      4. fabs-sqr 10.0%

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

      5. rem-square-sqrt 22.1%

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

   5. Simplified 22.1%

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

   6. Taylor expanded in x around 0 98.5%

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

      1. distribute-rgt-in 98.6%

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

      2. *-lft-identity 98.6%

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

      3. associate-*l* 98.6%

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

      4. unpow2 98.6%

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

      5. unpow3 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in x around 0 96.2%

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

   if 0.200000003 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. rem-square-sqrt 98.3%

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

      2. fabs-sqr 98.3%

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

      3. rem-square-sqrt 98.3%

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

   5. Simplified 98.3%

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

      1. *-un-lft-identity 98.3%

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

      2. log-prod 98.3%

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

      3. metadata-eval 98.3%

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

      4. *-inverses 98.3%

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

      5. metadata-eval 98.3%

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

   7. Applied egg-rr 98.3%

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

      1. +-lft-identity 98.3%

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

   9. Simplified 98.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.20000000298023224) (copysign x x) (copysign (log (* x 2.0)) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.200000003

   1. Initial program 32.0%

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

   3. Taylor expanded in x around 0 20.9%

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

      1. +-commutative 20.9%

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

      2. fma-define 20.9%

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

      3. rem-square-sqrt 7.3%

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

      4. fabs-sqr 7.3%

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

      5. rem-square-sqrt 21.1%

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

   5. Simplified 21.1%

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

   6. Taylor expanded in x around 0 74.3%

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

      1. distribute-rgt-in 74.3%

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

      2. *-lft-identity 74.3%

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

      3. associate-*l* 74.3%

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

      4. unpow2 74.3%

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

      5. unpow3 74.3%

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

   8. Simplified 74.3%

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

   9. Taylor expanded in x around 0 73.4%

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

   if 0.200000003 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. rem-square-sqrt 98.3%

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

      2. fabs-sqr 98.3%

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

      3. rem-square-sqrt 98.3%

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

   5. Simplified 98.3%

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

      1. *-un-lft-identity 98.3%

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

      2. log-prod 98.3%

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

      3. metadata-eval 98.3%

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

      4. *-inverses 98.3%

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

      5. metadata-eval 98.3%

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

   7. Applied egg-rr 98.3%

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

      1. +-lft-identity 98.3%

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

   9. Simplified 98.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.20000000298023224) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.200000003

   1. Initial program 32.0%

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

   3. Taylor expanded in x around 0 20.9%

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

      1. +-commutative 20.9%

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

      2. fma-define 20.9%

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

      3. rem-square-sqrt 7.3%

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

      4. fabs-sqr 7.3%

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

      5. rem-square-sqrt 21.1%

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

   5. Simplified 21.1%

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

   6. Taylor expanded in x around 0 74.3%

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

      1. distribute-rgt-in 74.3%

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

      2. *-lft-identity 74.3%

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

      3. associate-*l* 74.3%

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

      4. unpow2 74.3%

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

      5. unpow3 74.3%

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

   8. Simplified 74.3%

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

   9. Taylor expanded in x around 0 73.4%

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

   if 0.200000003 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around 0 44.5%

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

      1. log1p-define 44.5%

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

      2. rem-square-sqrt 44.5%

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

      3. fabs-sqr 44.5%

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

      4. rem-square-sqrt 44.5%

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

   5. Simplified 44.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.1% 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 38.8%

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

3. Taylor expanded in x around 0 20.2%

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

   1. +-commutative 20.2%

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

   2. fma-define 20.2%

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

   3. rem-square-sqrt 10.3%

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

   4. fabs-sqr 10.3%

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

   5. rem-square-sqrt 20.3%

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

5. Simplified 20.3%

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

6. Taylor expanded in x around 0 55.8%

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

   1. distribute-rgt-in 55.8%

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

   2. *-lft-identity 55.8%

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

   3. associate-*l* 55.8%

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

   4. unpow2 55.8%

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

   5. unpow3 55.8%

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

8. Simplified 55.8%

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

9. Taylor expanded in x around 0 56.1%

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

10. Final simplification 56.1%

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

Developer target: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024066 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

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