Rust f32::asinh

Percentage Accurate: 37.5% → 99.6%

Time: 6.9s

Alternatives: 14

Speedup: 3.8×

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: 37.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.5)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (+
         (* -0.16666666666666666 (pow x 3.0))
         (+ (* 0.075 (pow x 5.0)) (+ x (* -0.044642857142857144 (pow x 7.0)))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.5f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + ((0.075f * powf(x, 5.0f)) + (x + (-0.044642857142857144f * powf(x, 7.0f))))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.5))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(Float32(Float32(0.075) * (x ^ Float32(5.0))) + Float32(x + Float32(Float32(-0.044642857142857144) * (x ^ Float32(7.0)))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.5))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = sign(x) * abs(((single(-0.16666666666666666) * (x ^ single(3.0))) + ((single(0.075) * (x ^ single(5.0))) + (x + (single(-0.044642857142857144) * (x ^ single(7.0)))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 11.3%

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

      5. flip-+ 10.1%

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

      6. log-div 10.1%

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

      7. add-sqr-sqrt 10.2%

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

      8. fma-def 10.6%

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

      9. +-commutative 10.6%

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

      10. hypot-1-def 10.6%

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

   3. Applied egg-rr 10.6%

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

      1. fma-udef 10.2%

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

      2. +-commutative 10.2%

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

      3. associate--l+ 56.1%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. neg-sub0 100.0%

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

   5. Simplified 100.0%

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

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

   1. Initial program 20.5%

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

      1. *-un-lft-identity 20.5%

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

      2. *-commutative 20.5%

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

      3. log-prod 20.5%

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

      4. add-sqr-sqrt 10.5%

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

      5. fabs-sqr 10.5%

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

      6. add-sqr-sqrt 21.2%

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

      7. +-commutative 21.2%

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

      8. hypot-1-def 21.1%

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

      9. metadata-eval 21.1%

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

   3. Applied egg-rr 21.1%

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

      1. +-rgt-identity 21.1%

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

   5. Simplified 21.1%

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

   6. Taylor expanded in x around 0 100.0%

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

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

   1. Initial program 43.1%

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

      1. *-un-lft-identity 43.1%

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

      2. *-commutative 43.1%

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

      3. log-prod 43.1%

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

      4. add-sqr-sqrt 43.0%

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

      5. fabs-sqr 43.0%

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

      6. add-sqr-sqrt 43.1%

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

      7. +-commutative 43.1%

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

      8. hypot-1-def 98.7%

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

      9. metadata-eval 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. +-rgt-identity 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -0.10000000149011612f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + (x + (0.075f * powf(x, 5.0f)))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.10000000149011612))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(x + Float32(Float32(0.075) * (x ^ Float32(5.0))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-0.10000000149011612))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.10000000149011612))
		tmp = sign(x) * abs(((single(-0.16666666666666666) * (x ^ single(3.0))) + (x + (single(0.075) * (x ^ single(5.0))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 59.6%

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

      1. +-commutative 59.6%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 12.7%

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

      5. flip-+ 11.6%

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

      6. log-div 11.6%

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

      7. add-sqr-sqrt 11.6%

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

      8. fma-def 12.1%

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

      9. +-commutative 12.1%

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

      10. hypot-1-def 12.1%

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

   3. Applied egg-rr 12.1%

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

      1. fma-udef 11.6%

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

      2. +-commutative 11.6%

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

      3. associate--l+ 56.8%

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

      4. +-inverses 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. neg-sub0 99.9%

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

   5. Simplified 99.9%

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

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

   1. Initial program 19.3%

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

      1. *-un-lft-identity 19.3%

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

      2. *-commutative 19.3%

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

      3. log-prod 19.3%

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

      4. add-sqr-sqrt 9.9%

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

      5. fabs-sqr 9.9%

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

      6. add-sqr-sqrt 20.0%

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

      7. +-commutative 20.0%

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

      8. hypot-1-def 19.9%

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

      9. metadata-eval 19.9%

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

   3. Applied egg-rr 19.9%

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

      1. +-rgt-identity 19.9%

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

   5. Simplified 19.9%

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

   6. Taylor expanded in x around 0 100.0%

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

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

   1. Initial program 43.8%

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

      1. *-un-lft-identity 43.8%

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

      2. *-commutative 43.8%

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

      3. log-prod 43.8%

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

      4. add-sqr-sqrt 43.7%

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

      5. fabs-sqr 43.7%

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

      6. add-sqr-sqrt 43.8%

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

      7. +-commutative 43.8%

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

      8. hypot-1-def 98.6%

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

      9. metadata-eval 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-rgt-identity 98.6%

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

   5. 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 99.6%

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

Alternative 3: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 + 0.5 \cdot \frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (- (log (+ (* x -2.0) (* 0.5 (/ -1.0 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 <= -10.0f) {
		tmp = copysignf(-logf(((x * -2.0f) + (0.5f * (-1.0f / 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(-10.0))
		tmp = copysign(Float32(-log(Float32(Float32(x * Float32(-2.0)) + Float32(Float32(0.5) * Float32(Float32(-1.0) / 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(-10.0))
		tmp = sign(x) * abs(-log(((x * single(-2.0)) + (single(0.5) * (single(-1.0) / 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 < -10

   1. Initial program 58.2%

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

      1. +-commutative 58.2%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 9.7%

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

      5. flip-+ 8.6%

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

      6. log-div 8.6%

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

      7. add-sqr-sqrt 8.6%

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

      8. fma-def 9.1%

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

      9. +-commutative 9.1%

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

      10. hypot-1-def 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. fma-udef 8.6%

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

      2. +-commutative 8.6%

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

      3. associate--l+ 55.4%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. neg-sub0 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around -inf 99.8%

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

   if -10 < x < 0.100000001

   1. Initial program 20.5%

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

   2. Taylor expanded in x around 0 20.6%

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

      1. fma-def 20.6%

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

      2. unpow2 20.6%

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

      3. rem-square-sqrt 10.2%

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

      4. fabs-sqr 10.2%

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

      5. rem-square-sqrt 20.1%

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

      6. log1p-def 98.0%

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

      7. rem-square-sqrt 52.3%

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

      8. fabs-sqr 52.3%

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

      9. rem-square-sqrt 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in x around 0 99.2%

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

   if 0.100000001 < x

   1. Initial program 43.8%

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

      1. *-un-lft-identity 43.8%

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

      2. *-commutative 43.8%

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

      3. log-prod 43.8%

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

      4. add-sqr-sqrt 43.7%

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

      5. fabs-sqr 43.7%

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

      6. add-sqr-sqrt 43.8%

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

      7. +-commutative 43.8%

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

      8. hypot-1-def 98.6%

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

      9. metadata-eval 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-rgt-identity 98.6%

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

   5. 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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 + 0.5 \cdot \frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -0.100000001

   1. Initial program 59.6%

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

      1. +-commutative 59.6%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 12.7%

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

      5. flip-+ 11.6%

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

      6. log-div 11.6%

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

      7. add-sqr-sqrt 11.6%

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

      8. fma-def 12.1%

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

      9. +-commutative 12.1%

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

      10. hypot-1-def 12.1%

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

   3. Applied egg-rr 12.1%

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

      1. fma-udef 11.6%

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

      2. +-commutative 11.6%

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

      3. associate--l+ 56.8%

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

      4. +-inverses 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. neg-sub0 99.9%

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

   5. Simplified 99.9%

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

   if -0.100000001 < x < 0.100000001

   1. Initial program 19.3%

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

   2. Taylor expanded in x around 0 20.2%

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

      1. fma-def 20.2%

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

      2. unpow2 20.2%

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

      3. rem-square-sqrt 10.4%

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

      4. fabs-sqr 10.4%

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

      5. rem-square-sqrt 19.9%

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

      6. log1p-def 99.0%

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

      7. rem-square-sqrt 53.1%

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

      8. fabs-sqr 53.1%

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

      9. rem-square-sqrt 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in x around 0 99.9%

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

   if 0.100000001 < x

   1. Initial program 43.8%

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

      1. *-un-lft-identity 43.8%

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

      2. *-commutative 43.8%

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

      3. log-prod 43.8%

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

      4. add-sqr-sqrt 43.7%

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

      5. fabs-sqr 43.7%

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

      6. add-sqr-sqrt 43.8%

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

      7. +-commutative 43.8%

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

      8. hypot-1-def 98.6%

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

      9. metadata-eval 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-rgt-identity 98.6%

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

   5. 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 99.5%

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

Alternative 5: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 + 0.5 \cdot \frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (- (log (+ (* x -2.0) (* 0.5 (/ -1.0 x))))) x)
   (if (<= x 0.20000000298023224)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ (/ 0.5 x) (+ x x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(-logf(((x * -2.0f) + (0.5f * (-1.0f / x)))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (x + x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(Float32(-log(Float32(Float32(x * Float32(-2.0)) + Float32(Float32(0.5) * Float32(Float32(-1.0) / 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(Float32(Float32(0.5) / x) + Float32(x + x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(-log(((x * single(-2.0)) + (single(0.5) * (single(-1.0) / x)))));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (x + x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 58.2%

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

      1. +-commutative 58.2%

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

      2. add-sqr-sqrt -0.0%

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

      3. fabs-sqr -0.0%

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

      4. add-sqr-sqrt 9.7%

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

      5. flip-+ 8.6%

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

      6. log-div 8.6%

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

      7. add-sqr-sqrt 8.6%

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

      8. fma-def 9.1%

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

      9. +-commutative 9.1%

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

      10. hypot-1-def 9.1%

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

   3. Applied egg-rr 9.1%

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

      1. fma-udef 8.6%

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

      2. +-commutative 8.6%

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

      3. associate--l+ 55.4%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. neg-sub0 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around -inf 99.8%

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

   if -10 < x < 0.200000003

   1. Initial program 21.1%

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

   2. Taylor expanded in x around 0 20.9%

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

      1. fma-def 20.9%

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

      2. unpow2 20.9%

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

      3. rem-square-sqrt 10.6%

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

      4. fabs-sqr 10.6%

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

      5. rem-square-sqrt 20.4%

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

      6. log1p-def 97.6%

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

      7. rem-square-sqrt 52.3%

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

      8. fabs-sqr 52.3%

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

      9. rem-square-sqrt 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 99.0%

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

   if 0.200000003 < x

   1. Initial program 43.1%

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

   2. Taylor expanded in x around inf 97.3%

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

      1. associate-*r/ 97.3%

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

      2. metadata-eval 97.3%

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

      3. rem-square-sqrt 97.3%

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

      4. fabs-sqr 97.3%

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

      5. rem-square-sqrt 97.3%

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

   4. Simplified 97.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 + 0.5 \cdot \frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 6: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.20000000298023224)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ (/ 0.5 x) (+ x x))) x))))

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(x + x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		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(((single(0.5) / x) + (x + x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. *-commutative 58.2%

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

      3. log-prod 58.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 9.7%

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

      7. +-commutative 9.7%

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

      8. hypot-1-def 9.7%

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

      9. metadata-eval 9.7%

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

   3. Applied egg-rr 9.7%

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

      1. +-rgt-identity 9.7%

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

   5. Simplified 9.7%

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

   6. Taylor expanded in x around -inf 99.0%

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

   if -10 < x < 0.200000003

   1. Initial program 21.1%

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

   2. Taylor expanded in x around 0 20.9%

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

      1. fma-def 20.9%

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

      2. unpow2 20.9%

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

      3. rem-square-sqrt 10.6%

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

      4. fabs-sqr 10.6%

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

      5. rem-square-sqrt 20.4%

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

      6. log1p-def 97.6%

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

      7. rem-square-sqrt 52.3%

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

      8. fabs-sqr 52.3%

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

      9. rem-square-sqrt 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 99.0%

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

   if 0.200000003 < x

   1. Initial program 43.1%

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

   2. Taylor expanded in x around inf 97.3%

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

      1. associate-*r/ 97.3%

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

      2. metadata-eval 97.3%

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

      3. rem-square-sqrt 97.3%

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

      4. fabs-sqr 97.3%

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

      5. rem-square-sqrt 97.3%

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

   4. Simplified 97.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 7: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 2.0f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. *-commutative 58.2%

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

      3. log-prod 58.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 9.7%

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

      7. +-commutative 9.7%

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

      8. hypot-1-def 9.7%

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

      9. metadata-eval 9.7%

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

   3. Applied egg-rr 9.7%

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

      1. +-rgt-identity 9.7%

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

   5. Simplified 9.7%

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

   6. Taylor expanded in x around -inf 99.0%

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

   if -10 < x < 2

   1. Initial program 21.7%

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

   2. Taylor expanded in x around 0 21.1%

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

      1. fma-def 21.1%

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

      2. unpow2 21.1%

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

      3. rem-square-sqrt 10.8%

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

      4. fabs-sqr 10.8%

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

      5. rem-square-sqrt 20.5%

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

      6. log1p-def 97.2%

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

      7. rem-square-sqrt 52.2%

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

      8. fabs-sqr 52.2%

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

      9. rem-square-sqrt 97.8%

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

   4. Simplified 97.8%

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

   5. Taylor expanded in x around 0 98.5%

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

   if 2 < x

   1. Initial program 42.3%

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

   2. Taylor expanded in x around inf 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

      3. rem-square-sqrt 98.0%

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

      4. fabs-sqr 98.0%

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

      5. rem-square-sqrt 98.0%

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

   4. Simplified 98.0%

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

   5. Taylor expanded in x around 0 97.4%

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

4. Final simplification 98.3%

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

Alternative 8: 79.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -200.0)
   (copysign 15.333333333333334 x)
   (if (<= x 2.0) (copysign x x) (copysign (log (+ x x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -200.0f) {
		tmp = copysignf(15.333333333333334f, x);
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-200.0))
		tmp = copysign(Float32(15.333333333333334), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-200.0))
		tmp = sign(x) * abs(single(15.333333333333334));
	elseif (x <= single(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -200

   1. Initial program 57.5%

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

   2. Taylor expanded in x around inf -0.0%

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

      1. associate-*r/ -0.0%

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

      2. metadata-eval -0.0%

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

      3. rem-square-sqrt -0.0%

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

      4. fabs-sqr -0.0%

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

      5. rem-square-sqrt -0.0%

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

   4. Simplified -0.0%

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

   5. Taylor expanded in x around 0 -0.0%

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

   7. Simplified 27.9%

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

   if -200 < x < 2

   1. Initial program 22.3%

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

   2. Taylor expanded in x around 0 21.1%

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

      1. fma-def 21.1%

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

      2. unpow2 21.1%

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

      3. rem-square-sqrt 10.7%

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

      4. fabs-sqr 10.7%

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

      5. rem-square-sqrt 20.6%

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

      6. log1p-def 96.7%

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

      7. rem-square-sqrt 51.8%

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

      8. fabs-sqr 51.8%

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

      9. rem-square-sqrt 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in x around 0 97.9%

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

   6. Taylor expanded in x around 0 96.4%

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

   if 2 < x

   1. Initial program 42.3%

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

   2. Taylor expanded in x around inf 96.1%

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

      1. rem-square-sqrt 96.1%

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

      2. fabs-sqr 96.1%

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

      3. rem-square-sqrt 96.1%

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

   4. Simplified 96.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 9: 97.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 2.0) (copysign x x) (copysign (log (+ x x)) x))))

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. *-commutative 58.2%

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

      3. log-prod 58.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 9.7%

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

      7. +-commutative 9.7%

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

      8. hypot-1-def 9.7%

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

      9. metadata-eval 9.7%

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

   3. Applied egg-rr 9.7%

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

      1. +-rgt-identity 9.7%

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

   5. Simplified 9.7%

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

   6. Taylor expanded in x around -inf 99.0%

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

   if -10 < x < 2

   1. Initial program 21.7%

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

   2. Taylor expanded in x around 0 21.1%

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

      1. fma-def 21.1%

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

      2. unpow2 21.1%

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

      3. rem-square-sqrt 10.8%

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

      4. fabs-sqr 10.8%

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

      5. rem-square-sqrt 20.5%

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

      6. log1p-def 97.2%

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

      7. rem-square-sqrt 52.2%

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

      8. fabs-sqr 52.2%

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

      9. rem-square-sqrt 97.8%

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

   4. Simplified 97.8%

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

   5. Taylor expanded in x around 0 98.5%

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

   6. Taylor expanded in x around 0 97.0%

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

   if 2 < x

   1. Initial program 42.3%

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

   2. Taylor expanded in x around inf 96.1%

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

      1. rem-square-sqrt 96.1%

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

      2. fabs-sqr 96.1%

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

      3. rem-square-sqrt 96.1%

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

   4. Simplified 96.1%

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

4. Final simplification 97.2%

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

Alternative 10: 97.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 2.0) (copysign x x) (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. *-commutative 58.2%

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

      3. log-prod 58.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 9.7%

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

      7. +-commutative 9.7%

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

      8. hypot-1-def 9.7%

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

      9. metadata-eval 9.7%

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

   3. Applied egg-rr 9.7%

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

      1. +-rgt-identity 9.7%

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

   5. Simplified 9.7%

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

   6. Taylor expanded in x around -inf 99.0%

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

   if -10 < x < 2

   1. Initial program 21.7%

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

   2. Taylor expanded in x around 0 21.1%

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

      1. fma-def 21.1%

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

      2. unpow2 21.1%

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

      3. rem-square-sqrt 10.8%

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

      4. fabs-sqr 10.8%

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

      5. rem-square-sqrt 20.5%

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

      6. log1p-def 97.2%

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

      7. rem-square-sqrt 52.2%

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

      8. fabs-sqr 52.2%

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

      9. rem-square-sqrt 97.8%

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

   4. Simplified 97.8%

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

   5. Taylor expanded in x around 0 98.5%

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

   6. Taylor expanded in x around 0 97.0%

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

   if 2 < x

   1. Initial program 42.3%

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

   2. Taylor expanded in x around inf 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

      3. rem-square-sqrt 98.0%

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

      4. fabs-sqr 98.0%

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

      5. rem-square-sqrt 98.0%

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

   4. Simplified 98.0%

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

   5. Taylor expanded in x around 0 97.4%

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

4. Final simplification 97.6%

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

Alternative 11: 64.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < -10

   1. Initial program 58.2%

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

   2. Taylor expanded in x around inf -0.0%

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

      1. associate-*r/ -0.0%

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

      2. metadata-eval -0.0%

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

      3. rem-square-sqrt -0.0%

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

      4. fabs-sqr -0.0%

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

      5. rem-square-sqrt -0.0%

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

   4. Simplified -0.0%

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

   5. Taylor expanded in x around 0 -0.0%

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

   7. Simplified 27.8%

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

   if -10 < x

   1. Initial program 29.2%

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

   2. Taylor expanded in x around 0 27.8%

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

      1. log1p-def 75.8%

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

      2. rem-square-sqrt 48.2%

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

      3. fabs-sqr 48.2%

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

      4. rem-square-sqrt 75.7%

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

   4. Simplified 75.7%

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

4. Final simplification 65.2%

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

Alternative 12: 61.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{elif}\;x \leq 10:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -200.0)
   (copysign 15.333333333333334 x)
   (if (<= x 10.0) (copysign x x) (copysign 15.333333333333334 x))))

C

float code(float x) {
	float tmp;
	if (x <= -200.0f) {
		tmp = copysignf(15.333333333333334f, x);
	} else if (x <= 10.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(15.333333333333334f, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-200.0))
		tmp = copysign(Float32(15.333333333333334), x);
	elseif (x <= Float32(10.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(15.333333333333334), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-200.0))
		tmp = sign(x) * abs(single(15.333333333333334));
	elseif (x <= single(10.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(single(15.333333333333334));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -200 or 10 < x

   1. Initial program 48.0%

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

   2. Taylor expanded in x around inf 55.6%

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

      1. associate-*r/ 55.6%

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

      2. metadata-eval 55.6%

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

      3. rem-square-sqrt 55.6%

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

      4. fabs-sqr 55.6%

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

      5. rem-square-sqrt 55.6%

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

   4. Simplified 55.6%

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

   5. Taylor expanded in x around 0 0.8%

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

   7. Simplified 27.1%

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

   if -200 < x < 10

   1. Initial program 23.5%

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

   2. Taylor expanded in x around 0 21.2%

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

      1. fma-def 21.2%

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

      2. unpow2 21.2%

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

      3. rem-square-sqrt 11.0%

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

      4. fabs-sqr 11.0%

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

      5. rem-square-sqrt 20.8%

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

      6. log1p-def 95.6%

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

      7. rem-square-sqrt 51.4%

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

      8. fabs-sqr 51.4%

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

      9. rem-square-sqrt 96.0%

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

   4. Simplified 96.0%

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

   5. Taylor expanded in x around 0 96.7%

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

   6. Taylor expanded in x around 0 95.4%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \mathbf{elif}\;x \leq 10:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(15.333333333333334, x\right)\\ \end{array} \]

Alternative 13: 17.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 35.6%

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

2. Taylor expanded in x around inf 30.8%

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

   1. associate-*r/ 30.8%

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

   2. metadata-eval 30.8%

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

   3. rem-square-sqrt 30.8%

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

   4. fabs-sqr 30.8%

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

   5. rem-square-sqrt 30.8%

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

4. Simplified 30.8%

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

5. Taylor expanded in x around 0 3.7%

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

7. Simplified 17.4%

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

8. Final simplification 17.4%

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

Alternative 14: 18.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 35.6%

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

2. Taylor expanded in x around inf 30.8%

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

   1. associate-*r/ 30.8%

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

   2. metadata-eval 30.8%

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

   3. rem-square-sqrt 30.8%

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

   4. fabs-sqr 30.8%

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

   5. rem-square-sqrt 30.8%

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

4. Simplified 30.8%

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

5. Taylor expanded in x around 0 3.3%

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

7. Simplified 18.7%

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

8. Final simplification 18.7%

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

Developer target: 99.7% 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 2023257 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

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