Rust f32::asinh

Percentage Accurate: 38.2% → 99.5%

Time: 10.4s

Alternatives: 12

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 38.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + \mathsf{hypot}\left(1, x\right)\\ t_1 := \log \left({t\_0}^{-0.5}\right)\\ t_2 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_2 \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\left(-t\_1\right) - t\_1, x\right)\\ \mathbf{elif}\;t\_2 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log t\_0, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ (fabs x) (hypot 1.0 x)))
        (t_1 (log (pow t_0 -0.5)))
        (t_2 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_2 -0.10000000149011612)
     (copysign (- (- t_1) t_1) x)
     (if (<= t_2 0.10000000149011612)
       (copysign (log1p (+ (fabs x) (* (* x x) (+ 0.5 (* (* x x) -0.125))))) x)
       (copysign (log t_0) x)))))

C

float code(float x) {
	float t_0 = fabsf(x) + hypotf(1.0f, x);
	float t_1 = logf(powf(t_0, -0.5f));
	float t_2 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_2 <= -0.10000000149011612f) {
		tmp = copysignf((-t_1 - t_1), x);
	} else if (t_2 <= 0.10000000149011612f) {
		tmp = copysignf(log1pf((fabsf(x) + ((x * x) * (0.5f + ((x * x) * -0.125f))))), x);
	} else {
		tmp = copysignf(logf(t_0), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(abs(x) + hypot(Float32(1.0), x))
	t_1 = log((t_0 ^ Float32(-0.5)))
	t_2 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_2 <= Float32(-0.10000000149011612))
		tmp = copysign(Float32(Float32(-t_1) - t_1), x);
	elseif (t_2 <= Float32(0.10000000149011612))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(Float32(x * x) * Float32(-0.125)))))), x);
	else
		tmp = copysign(log(t_0), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{1 + x \cdot x}\right), x\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right), x\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{{\left(\left|x\right|\right)}^{3} + {\left(\sqrt{1 \cdot 1 + x \cdot x}\right)}^{3}}{\left|x\right| \cdot \left|x\right| + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)}\right), x\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{1}{\frac{\left|x\right| \cdot \left|x\right| + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)}{{\left(\left|x\right|\right)}^{3} + {\left(\sqrt{1 \cdot 1 + x \cdot x}\right)}^{3}}}\right), x\right) \]

      5. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)}{{\left(\left|x\right|\right)}^{3} + {\left(\sqrt{1 \cdot 1 + x \cdot x}\right)}^{3}}\right)\right)\right), x\right) \]

      6. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)}{{\left(\left|x\right|\right)}^{3} + {\left(\sqrt{1 \cdot 1 + x \cdot x}\right)}^{3}}\right)\right), x\right) \]

   4. Applied egg-rr 99.8%

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

      1. inv-pow N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{-1}\right)\right), x\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right), x\right) \]

      3. log-prod N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\left(\log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right) + \log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{+.f32}\left(\log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right), \log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right), x\right) \]

      5. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right), \log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\frac{-1}{2}}\right)\right), \log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right), x\right) \]

      7. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{pow.f32}\left(\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right), \frac{-1}{2}\right)\right), \log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right), x\right) \]

      8. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right), \frac{-1}{2}\right)\right), \log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right), x\right) \]

      9. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right), \frac{-1}{2}\right)\right), \log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right), x\right) \]

      10. hypot-undefine N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right), \frac{-1}{2}\right)\right), \log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right), x\right) \]

      11. hypot-lowering-hypot.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right), \frac{-1}{2}\right)\right), \log \left({\left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right), x\right) \]

   6. Applied egg-rr 99.9%

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

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

   1. Initial program 21.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 + \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right| + 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      5. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(x \cdot \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      16. *-lowering-*.f32 21.4%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.4%

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

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + 1\right) + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right), \color{blue}{x}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(1 + \left|x\right|\right) + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(1 + \left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      5. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      6. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      7. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      13. *-lowering-*.f32 99.9%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)} \]

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

   1. Initial program 52.4%

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

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\left(-\log \left({\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{-0.5}\right)\right) - \log \left({\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{-0.5}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + \mathsf{hypot}\left(1, x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_1 \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{t\_0}\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log t\_0, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ (fabs x) (hypot 1.0 x)))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_1 -0.10000000149011612)
     (copysign (- (log (/ 1.0 t_0))) x)
     (if (<= t_1 0.10000000149011612)
       (copysign (log1p (+ (fabs x) (* (* x x) (+ 0.5 (* (* x x) -0.125))))) x)
       (copysign (log t_0) x)))))

C

float code(float x) {
	float t_0 = fabsf(x) + hypotf(1.0f, x);
	float t_1 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_1 <= -0.10000000149011612f) {
		tmp = copysignf(-logf((1.0f / t_0)), x);
	} else if (t_1 <= 0.10000000149011612f) {
		tmp = copysignf(log1pf((fabsf(x) + ((x * x) * (0.5f + ((x * x) * -0.125f))))), x);
	} else {
		tmp = copysignf(logf(t_0), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(abs(x) + hypot(Float32(1.0), x))
	t_1 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_1 <= Float32(-0.10000000149011612))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / t_0))), x);
	elseif (t_1 <= Float32(0.10000000149011612))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(Float32(x * x) * Float32(-0.125)))))), x);
	else
		tmp = copysign(log(t_0), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{1 + x \cdot x}\right), x\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{1 \cdot 1 + x \cdot x}\right), x\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{{\left(\left|x\right|\right)}^{3} + {\left(\sqrt{1 \cdot 1 + x \cdot x}\right)}^{3}}{\left|x\right| \cdot \left|x\right| + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)}\right), x\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{1}{\frac{\left|x\right| \cdot \left|x\right| + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)}{{\left(\left|x\right|\right)}^{3} + {\left(\sqrt{1 \cdot 1 + x \cdot x}\right)}^{3}}}\right), x\right) \]

      5. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)}{{\left(\left|x\right|\right)}^{3} + {\left(\sqrt{1 \cdot 1 + x \cdot x}\right)}^{3}}\right)\right)\right), x\right) \]

      6. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\frac{\left|x\right| \cdot \left|x\right| + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)}{{\left(\left|x\right|\right)}^{3} + {\left(\sqrt{1 \cdot 1 + x \cdot x}\right)}^{3}}\right)\right), x\right) \]

   4. Applied egg-rr 99.8%

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

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

   1. Initial program 21.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 + \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right| + 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      5. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(x \cdot \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      16. *-lowering-*.f32 21.4%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.4%

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

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + 1\right) + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right), \color{blue}{x}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(1 + \left|x\right|\right) + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(1 + \left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      5. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      6. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      7. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      13. *-lowering-*.f32 99.9%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)} \]

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

   1. Initial program 52.4%

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

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 100.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

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

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.10000000149011612:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))
   (if (<= t_0 -0.10000000149011612)
     t_1
     (if (<= t_0 0.10000000149011612)
       (copysign (log1p (+ (fabs x) (* (* x x) (+ 0.5 (* (* x x) -0.125))))) x)
       t_1))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	float tmp;
	if (t_0 <= -0.10000000149011612f) {
		tmp = t_1;
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = copysignf(log1pf((fabsf(x) + ((x * x) * (0.5f + ((x * x) * -0.125f))))), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.10000000149011612))
		tmp = t_1;
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(Float32(x * x) * Float32(-0.125)))))), x);
	else
		tmp = t_1;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 53.8%

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

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f32 99.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{hypot.f32}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 99.9%

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

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

   1. Initial program 21.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 + \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right| + 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      5. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(x \cdot \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      16. *-lowering-*.f32 21.4%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.4%

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

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + 1\right) + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right), \color{blue}{x}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(1 + \left|x\right|\right) + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(1 + \left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      5. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      6. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      7. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      13. *-lowering-*.f32 99.9%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right) + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- (+ (fabs x) (/ -0.5 x)) x)) x)
   (if (<= x 0.4000000059604645)
     (copysign (log1p (+ (fabs x) (* (* x x) (+ 0.5 (* (* x x) -0.125))))) x)
     (copysign
      (log (+ (+ (fabs x) (+ x (/ 0.5 x))) (/ -0.125 (* x (* x x)))))
      x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) + (-0.5f / x)) - x)), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf(log1pf((fabsf(x) + ((x * x) * (0.5f + ((x * x) * -0.125f))))), x);
	} else {
		tmp = copysignf(logf(((fabsf(x) + (x + (0.5f / x))) + (-0.125f / (x * (x * x))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) + Float32(Float32(-0.5) / x)) - x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(Float32(x * x) * Float32(-0.125)))))), x);
	else
		tmp = copysign(log(Float32(Float32(abs(x) + Float32(x + Float32(Float32(0.5) / x))) + Float32(Float32(-0.125) / Float32(x * Float32(x * x))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   5. Simplified 52.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\color{blue}{\left(\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}\right)}, x\right)\right), x\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\left|x\right| + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right), x\right) \]

      8. /-lowering-/.f32 99.3%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right) \]

   8. Simplified 99.3%

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

   if -2 < x < 0.400000006

   1. Initial program 23.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 + \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right| + 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      5. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(x \cdot \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      16. *-lowering-*.f32 22.9%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 22.9%

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

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + 1\right) + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right), \color{blue}{x}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(1 + \left|x\right|\right) + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(1 + \left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      5. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      6. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      7. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      13. *-lowering-*.f32 98.9%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)} \]

   if 0.400000006 < x

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \color{blue}{\left(x \cdot \left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{\left(2 \cdot 2\right)}}\right)\right)\right)\right)\right), x\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{2} \cdot {x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(\frac{\frac{-1}{8}}{{x}^{2} \cdot {x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. cube-mult N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left(x \cdot {x}^{3}\right)\right)\right)\right)\right)\right), x\right) \]

      19. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(x, \left({x}^{3}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      20. cube-mult N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(x, \left(x \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      22. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      24. *-lowering-*.f32 100.0%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(x + \left(\left|x\right| + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right), x\right)} \]
   7. Step-by-step derivation

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(x + \left(\left|x\right| + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right), \color{blue}{x}\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right) + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- (+ (fabs x) (/ -0.5 x)) x)) x)
   (if (<= x 0.4000000059604645)
     (copysign (log1p (+ (fabs x) (* (* x x) (+ 0.5 (* (* x x) -0.125))))) x)
     (copysign (log (+ (/ 0.5 x) (+ x (fabs x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) + (-0.5f / x)) - x)), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf(log1pf((fabsf(x) + ((x * x) * (0.5f + ((x * x) * -0.125f))))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (x + fabsf(x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) + Float32(Float32(-0.5) / x)) - x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(Float32(x * x) * Float32(-0.125)))))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(x + abs(x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   5. Simplified 52.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\color{blue}{\left(\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}\right)}, x\right)\right), x\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\left|x\right| + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right), x\right) \]

      8. /-lowering-/.f32 99.3%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right) \]

   8. Simplified 99.3%

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

   if -2 < x < 0.400000006

   1. Initial program 23.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 + \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right| + 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      5. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \left(x \cdot \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right), x\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{8} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left({x}^{2} \cdot \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

      16. *-lowering-*.f32 22.9%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), 1\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{8}\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 22.9%

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

      1. copysign-lowering-copysign.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(\left|x\right| + 1\right) + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right), \color{blue}{x}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\left(1 + \left|x\right|\right) + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(1 + \left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      5. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      6. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      7. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right), x\right) \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

      13. *-lowering-*.f32 98.9%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{8}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)} \]

   if 0.400000006 < x

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f32 99.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 99.6%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \left(1 \cdot x + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \left(x + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\left|x\right| + x\right) + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right| + x\right), \left(\frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), x\right), \left(\frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      6. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{x \cdot x}\right) \cdot x\right)\right)\right), x\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot x\right)\right)\right)\right), x\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot x\right)\right)\right)\right), x\right) \]

      10. pow-flip N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \left({x}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot x\right)\right)\right)\right), x\right) \]

      11. pow-plus N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) + 1\right)}\right)\right)\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot {x}^{\left(-2 + 1\right)}\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot {x}^{-1}\right)\right)\right), x\right) \]

      14. inv-pow N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right) \]

      16. /-lowering-/.f32 99.6%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- (+ (fabs x) (/ -0.5 x)) x)) x)
   (if (<= x 0.4000000059604645)
     (copysign x x)
     (copysign (log (+ (/ 0.5 x) (+ x (fabs x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) + (-0.5f / x)) - x)), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (x + fabsf(x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) + Float32(Float32(-0.5) / x)) - x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(x + abs(x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log(((abs(x) + (single(-0.5) / x)) - x)));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (x + abs(x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   5. Simplified 52.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\color{blue}{\left(\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}\right)}, x\right)\right), x\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\left|x\right| + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right), x\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right), x\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right), x\right) \]

      8. /-lowering-/.f32 99.3%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right)\right), x\right) \]

   8. Simplified 99.3%

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

   if -2 < x < 0.400000006

   1. Initial program 23.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 93.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 93.0%

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

   7. Applied egg-rr 14.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 97.1%

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

   if 0.400000006 < x

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f32 99.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 99.6%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \left(1 \cdot x + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \left(x + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\left|x\right| + x\right) + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right| + x\right), \left(\frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), x\right), \left(\frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      6. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{x \cdot x}\right) \cdot x\right)\right)\right), x\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot x\right)\right)\right)\right), x\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot x\right)\right)\right)\right), x\right) \]

      10. pow-flip N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \left({x}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot x\right)\right)\right)\right), x\right) \]

      11. pow-plus N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) + 1\right)}\right)\right)\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot {x}^{\left(-2 + 1\right)}\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot {x}^{-1}\right)\right)\right), x\right) \]

      14. inv-pow N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right) \]

      16. /-lowering-/.f32 99.6%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.4000000059604645)
     (copysign x x)
     (copysign (log (+ (/ 0.5 x) (+ x (fabs x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (x + fabsf(x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(x + abs(x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (x + abs(x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   5. Simplified 52.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 97.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right) \]

   8. Simplified 97.7%

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

   if -2 < x < 0.400000006

   1. Initial program 23.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 93.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 93.0%

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

   7. Applied egg-rr 14.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 97.1%

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

   if 0.400000006 < x

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f32 99.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 99.6%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \left(1 \cdot x + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \left(x + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\left|x\right| + x\right) + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right), x\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right| + x\right), \left(\frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), x\right), \left(\frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      6. fabs-lowering-fabs.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{x \cdot x}\right) \cdot x\right)\right)\right), x\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot x\right)\right)\right)\right), x\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot x\right)\right)\right)\right), x\right) \]

      10. pow-flip N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \left({x}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot x\right)\right)\right)\right), x\right) \]

      11. pow-plus N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) + 1\right)}\right)\right)\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot {x}^{\left(-2 + 1\right)}\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot {x}^{-1}\right)\right)\right), x\right) \]

      14. inv-pow N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right) \]

      16. /-lowering-/.f32 99.6%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), x\right), \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.4000000059604645) (copysign x x) (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.4000000059604645f) {
		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(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.4000000059604645))
		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(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.4000000059604645))
		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 < -2

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   5. Simplified 52.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 97.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right) \]

   8. Simplified 97.7%

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

   if -2 < x < 0.400000006

   1. Initial program 23.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 93.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 93.0%

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

   7. Applied egg-rr 14.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 97.1%

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

   if 0.400000006 < x

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f32 99.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 98.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), x\right) \]

   8. Simplified 98.2%

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

Alternative 9: 83.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.4000000059604645) (copysign x x) (copysign (log x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), x\right) \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - \left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right) - x\right)\right), x\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}}{x}\right) \cdot x\right)\right), x\right)\right), x\right) \]

   5. Simplified 52.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 97.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{-1}{2}, x\right)\right), x\right) \]

   8. Simplified 97.7%

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

   if -2 < x < 0.400000006

   1. Initial program 23.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 93.0%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 93.0%

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

   7. Applied egg-rr 14.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 97.1%

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

   if 0.400000006 < x

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log x, x\right) \]

      4. log-lowering-log.f32 44.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   5. Simplified 44.7%

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

Alternative 10: 70.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -20.0)
   (copysign (log (- x)) x)
   (if (<= x 0.4000000059604645) (copysign x x) (copysign (log x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -20.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.4000000059604645f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-20.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.4000000059604645))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-20.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(0.4000000059604645))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -20

   1. Initial program 52.5%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f32 44.5%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{neg.f32}\left(x\right)\right), x\right) \]

   5. Simplified 44.5%

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

   if -20 < x < 0.400000006

   1. Initial program 24.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 92.5%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 92.5%

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

   7. Applied egg-rr 14.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 96.6%

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

   if 0.400000006 < x

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log x, x\right) \]

      4. log-lowering-log.f32 44.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   5. Simplified 44.7%

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

Alternative 11: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.4000000059604645:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.4000000059604645) (copysign x x) (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.4000000059604645f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.4000000059604645))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(0.4000000059604645))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.400000006

   1. Initial program 33.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f32 76.4%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 76.4%

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

   7. Applied egg-rr 9.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 67.8%

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

   if 0.400000006 < x

   1. Initial program 51.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log x, x\right) \]

      4. log-lowering-log.f32 44.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   5. Simplified 44.7%

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

Alternative 12: 53.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 38.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
4. Step-by-step derivation

   1. log1p-define N/A

      \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

   2. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

   3. fabs-lowering-fabs.f32 68.8%

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

5. Simplified 68.8%

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

7. Applied egg-rr 7.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
9. Step-by-step derivation

10. Simplified 54.0%

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

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024144 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

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

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