Rust f32::asinh

Percentage Accurate: 37.7% → 99.3%

Time: 9.5s

Alternatives: 15

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.7% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.004999999888241291)
       (copysign
        (+
         x
         (*
          x
          (*
           (* x x)
           (+
            (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
            -0.16666666666666666))))
        x)
       (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.004999999888241291f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.004999999888241291))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.004999999888241291))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((abs(x) + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

      1. flip-+ 12.5%

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

      2. log-div 12.5%

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

      3. hypot-1-def 12.5%

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

      4. hypot-1-def 12.5%

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

      5. add-sqr-sqrt 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. associate--l+ 52.4%

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

      2. +-inverses 99.9%

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

      3. metadata-eval 99.9%

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

      4. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.00499999989

   1. Initial program 20.7%

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

   3. Taylor expanded in x around 0 20.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.1%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.1%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.1%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. unpow2 100.0%

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

      5. *-commutative 100.0%

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

      6. unpow2 100.0%

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

      7. metadata-eval 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 0.00499999989 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 61.1%

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

      1. +-commutative 61.1%

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

      2. hypot-1-def 98.6%

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

   3. Simplified 98.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.004999999888241291)
     (copysign
      (+
       x
       (*
        x
        (*
         (* x x)
         (+
          (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
          -0.16666666666666666))))
      x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.004999999888241291f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.004999999888241291))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.004999999888241291))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

      1. flip-+ 12.5%

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

      2. log-div 12.5%

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

      3. hypot-1-def 12.5%

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

      4. hypot-1-def 12.5%

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

      5. add-sqr-sqrt 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. associate--l+ 52.4%

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

      2. +-inverses 99.9%

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

      3. metadata-eval 99.9%

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

      4. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   if -2 < x < 0.00499999989

   1. Initial program 20.7%

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

   3. Taylor expanded in x around 0 20.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.1%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.1%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.1%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. unpow2 100.0%

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

      5. *-commutative 100.0%

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

      6. unpow2 100.0%

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

      7. metadata-eval 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 0.00499999989 < x

   1. Initial program 61.1%

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

      1. +-commutative 61.1%

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

      2. +-commutative 61.1%

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

      3. hypot-1-def 98.6%

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

      4. add-sqr-sqrt 98.6%

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

      5. fabs-sqr 98.6%

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

      6. add-sqr-sqrt 98.6%

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

   4. Applied egg-rr 98.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\left(\left(\frac{0.125}{x \cdot x} + \frac{0.0390625}{t\_0 \cdot t\_0}\right) - \frac{0.0625}{x \cdot t\_0}\right) - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (* x (* x x))))
   (if (<= x -2.0)
     (copysign
      (log
       (/
        (-
         (-
          (+ (/ 0.125 (* x x)) (/ 0.0390625 (* t_0 t_0)))
          (/ 0.0625 (* x t_0)))
         0.5)
        x))
      x)
     (if (<= x 0.004999999888241291)
       (copysign
        (+
         x
         (*
          x
          (*
           (* x x)
           (+
            (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
            -0.16666666666666666))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

float code(float x) {
	float t_0 = x * (x * x);
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((((((0.125f / (x * x)) + (0.0390625f / (t_0 * t_0))) - (0.0625f / (x * t_0))) - 0.5f) / x)), x);
	} else if (x <= 0.004999999888241291f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(x * Float32(x * x))
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(Float32(Float32(Float32(Float32(0.125) / Float32(x * x)) + Float32(Float32(0.0390625) / Float32(t_0 * t_0))) - Float32(Float32(0.0625) / Float32(x * t_0))) - Float32(0.5)) / x)), x);
	elseif (x <= Float32(0.004999999888241291))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((((((single(0.125) / (x * x)) + (single(0.0390625) / (t_0 * t_0))) - (single(0.0625) / (x * t_0))) - single(0.5)) / x)));
	elseif (x <= single(0.004999999888241291))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in x around -inf 99.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - \left(0.125 \cdot \frac{1}{{x}^{2}} + 0.0390625 \cdot \frac{1}{{x}^{6}}\right)}{x}\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - \left(0.125 \cdot \frac{1}{{x}^{2}} + 0.0390625 \cdot \frac{1}{{x}^{6}}\right)}{x}\right)}, x\right) \]

      2. neg-sub0 99.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - \left(0.125 \cdot \frac{1}{{x}^{2}} + 0.0390625 \cdot \frac{1}{{x}^{6}}\right)}{x}\right)}, x\right) \]

   7. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \frac{0.5 + \left(\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} - \left(\frac{0.125}{x \cdot x} + \frac{0.0390625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{x}\right)}, x\right) \]

   if -2 < x < 0.00499999989

   1. Initial program 20.7%

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

   3. Taylor expanded in x around 0 20.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.1%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.1%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.1%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. unpow2 100.0%

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

      5. *-commutative 100.0%

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

      6. unpow2 100.0%

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

      7. metadata-eval 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 0.00499999989 < x

   1. Initial program 61.1%

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

      1. +-commutative 61.1%

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

      2. +-commutative 61.1%

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

      3. hypot-1-def 98.6%

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

      4. add-sqr-sqrt 98.6%

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

      5. fabs-sqr 98.6%

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

      6. add-sqr-sqrt 98.6%

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

   4. Applied egg-rr 98.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\left(\left(\frac{0.125}{x \cdot x} + \frac{0.0390625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) - \frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := x \cdot t\_0\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\left(\left(\frac{0.125}{x \cdot x} + \frac{0.0390625}{t\_1}\right) - \frac{0.0625}{t\_2}\right) - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{t\_1} + \frac{0.5}{x \cdot x}\right) + \frac{-0.125}{t\_2}\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (* x (* x x))) (t_1 (* t_0 t_0)) (t_2 (* x t_0)))
   (if (<= x -2.0)
     (copysign
      (log
       (/
        (- (- (+ (/ 0.125 (* x x)) (/ 0.0390625 t_1)) (/ 0.0625 t_2)) 0.5)
        x))
      x)
     (if (<= x 1.0)
       (copysign
        (+
         x
         (*
          x
          (*
           (* x x)
           (+
            (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
            -0.16666666666666666))))
        x)
       (copysign
        (log
         (* x (+ 2.0 (+ (+ (/ 0.0625 t_1) (/ 0.5 (* x x))) (/ -0.125 t_2)))))
        x)))))

C

float code(float x) {
	float t_0 = x * (x * x);
	float t_1 = t_0 * t_0;
	float t_2 = x * t_0;
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((((((0.125f / (x * x)) + (0.0390625f / t_1)) - (0.0625f / t_2)) - 0.5f) / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x * (2.0f + (((0.0625f / t_1) + (0.5f / (x * x))) + (-0.125f / t_2))))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(x * Float32(x * x))
	t_1 = Float32(t_0 * t_0)
	t_2 = Float32(x * t_0)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(Float32(Float32(Float32(Float32(0.125) / Float32(x * x)) + Float32(Float32(0.0390625) / t_1)) - Float32(Float32(0.0625) / t_2)) - Float32(0.5)) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(0.0625) / t_1) + Float32(Float32(0.5) / Float32(x * x))) + Float32(Float32(-0.125) / t_2))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = t_0 * t_0;
	t_2 = x * t_0;
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((((((single(0.125) / (x * x)) + (single(0.0390625) / t_1)) - (single(0.0625) / t_2)) - single(0.5)) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x * (single(2.0) + (((single(0.0625) / t_1) + (single(0.5) / (x * x))) + (single(-0.125) / t_2))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in x around -inf 99.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - \left(0.125 \cdot \frac{1}{{x}^{2}} + 0.0390625 \cdot \frac{1}{{x}^{6}}\right)}{x}\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - \left(0.125 \cdot \frac{1}{{x}^{2}} + 0.0390625 \cdot \frac{1}{{x}^{6}}\right)}{x}\right)}, x\right) \]

      2. neg-sub0 99.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - \left(0.125 \cdot \frac{1}{{x}^{2}} + 0.0390625 \cdot \frac{1}{{x}^{6}}\right)}{x}\right)}, x\right) \]

   7. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \frac{0.5 + \left(\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} - \left(\frac{0.125}{x \cdot x} + \frac{0.0390625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{x}\right)}, x\right) \]

   if -2 < x < 1

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 21.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. unpow2 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

      2. +-commutative 59.9%

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

      3. hypot-1-def 98.5%

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

      4. add-sqr-sqrt 98.5%

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

      5. fabs-sqr 98.5%

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

      6. add-sqr-sqrt 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in x around inf 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\left(2 + \left(0.5 \cdot \frac{1}{{x}^{2}} + 0.0625 \cdot \frac{1}{{x}^{6}}\right)\right) - \frac{0.125}{{x}^{4}}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. associate--l+ 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \color{blue}{\left(2 + \left(\left(0.5 \cdot \frac{1}{{x}^{2}} + 0.0625 \cdot \frac{1}{{x}^{6}}\right) - \frac{0.125}{{x}^{4}}\right)\right)}\right), x\right) \]

      2. sub-neg 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \color{blue}{\left(\left(0.5 \cdot \frac{1}{{x}^{2}} + 0.0625 \cdot \frac{1}{{x}^{6}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)}\right)\right), x\right) \]

      3. +-commutative 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\color{blue}{\left(0.0625 \cdot \frac{1}{{x}^{6}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)} + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      4. associate-*r/ 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\color{blue}{\frac{0.0625 \cdot 1}{{x}^{6}}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      5. metadata-eval 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{\color{blue}{0.0625}}{{x}^{6}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      6. metadata-eval 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{{x}^{\color{blue}{\left(2 \cdot 3\right)}}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      7. pow-sqr 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\color{blue}{{x}^{3} \cdot {x}^{3}}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      8. cube-unmult 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot {x}^{3}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      9. cube-unmult 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      10. unpow2 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + 0.5 \cdot \frac{1}{\color{blue}{x \cdot x}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      11. associate-*r/ 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \color{blue}{\frac{0.5 \cdot 1}{x \cdot x}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      12. metadata-eval 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\color{blue}{0.5}}{x \cdot x}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      13. metadata-eval 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \left(-\frac{0.125}{{x}^{\color{blue}{\left(3 + 1\right)}}}\right)\right)\right)\right), x\right) \]

      14. pow-plus 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \left(-\frac{0.125}{\color{blue}{{x}^{3} \cdot x}}\right)\right)\right)\right), x\right) \]

      15. unpow3 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \left(-\frac{0.125}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot x}\right)\right)\right)\right), x\right) \]

      16. associate-*r* 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \left(-\frac{0.125}{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}\right)\right)\right)\right), x\right) \]

   7. Simplified 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\left(\left(\frac{0.125}{x \cdot x} + \frac{0.0390625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) - \frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\left(\frac{0.125}{x \cdot x} - 0.5\right) - \frac{0.0625}{t\_1}}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{t\_0 \cdot t\_0} + \frac{0.5}{x \cdot x}\right) + \frac{-0.125}{t\_1}\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (* x (* x x))) (t_1 (* x t_0)))
   (if (<= x -2.0)
     (copysign (log (/ (- (- (/ 0.125 (* x x)) 0.5) (/ 0.0625 t_1)) x)) x)
     (if (<= x 1.0)
       (copysign
        (+
         x
         (*
          x
          (*
           (* x x)
           (+
            (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
            -0.16666666666666666))))
        x)
       (copysign
        (log
         (*
          x
          (+
           2.0
           (+ (+ (/ 0.0625 (* t_0 t_0)) (/ 0.5 (* x x))) (/ -0.125 t_1)))))
        x)))))

C

float code(float x) {
	float t_0 = x * (x * x);
	float t_1 = x * t_0;
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((((0.125f / (x * x)) - 0.5f) - (0.0625f / t_1)) / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x * (2.0f + (((0.0625f / (t_0 * t_0)) + (0.5f / (x * x))) + (-0.125f / t_1))))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(x * Float32(x * x))
	t_1 = Float32(x * t_0)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(Float32(Float32(Float32(0.125) / Float32(x * x)) - Float32(0.5)) - Float32(Float32(0.0625) / t_1)) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(2.0) + Float32(Float32(Float32(Float32(0.0625) / Float32(t_0 * t_0)) + Float32(Float32(0.5) / Float32(x * x))) + Float32(Float32(-0.125) / t_1))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = x * t_0;
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log(((((single(0.125) / (x * x)) - single(0.5)) - (single(0.0625) / t_1)) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x * (single(2.0) + (((single(0.0625) / (t_0 * t_0)) + (single(0.5) / (x * x))) + (single(-0.125) / t_1))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in x around -inf 99.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]

      2. neg-sub0 99.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]

      3. +-commutative 99.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\color{blue}{\left(\frac{0.0625}{{x}^{4}} + 0.5\right)} - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right), x\right) \]

      4. associate--l+ 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\color{blue}{\frac{0.0625}{{x}^{4}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}}{x}\right), x\right) \]

      5. metadata-eval 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{{x}^{\color{blue}{\left(3 + 1\right)}}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      6. pow-plus 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{{x}^{3} \cdot x}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      7. unpow3 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot x} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      8. associate-*r* 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      9. associate-*l* 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      10. unpow2 99.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - 0.125 \cdot \frac{1}{\color{blue}{x \cdot x}}\right)}{x}\right), x\right) \]

      11. associate-*r/ 99.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - \color{blue}{\frac{0.125 \cdot 1}{x \cdot x}}\right)}{x}\right), x\right) \]

      12. metadata-eval 99.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - \frac{\color{blue}{0.125}}{x \cdot x}\right)}{x}\right), x\right) \]

   7. Simplified 99.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - \frac{0.125}{x \cdot x}\right)}{x}\right)}, x\right) \]

   if -2 < x < 1

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 21.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. unpow2 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

      2. +-commutative 59.9%

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

      3. hypot-1-def 98.5%

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

      4. add-sqr-sqrt 98.5%

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

      5. fabs-sqr 98.5%

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

      6. add-sqr-sqrt 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in x around inf 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\left(2 + \left(0.5 \cdot \frac{1}{{x}^{2}} + 0.0625 \cdot \frac{1}{{x}^{6}}\right)\right) - \frac{0.125}{{x}^{4}}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. associate--l+ 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \color{blue}{\left(2 + \left(\left(0.5 \cdot \frac{1}{{x}^{2}} + 0.0625 \cdot \frac{1}{{x}^{6}}\right) - \frac{0.125}{{x}^{4}}\right)\right)}\right), x\right) \]

      2. sub-neg 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \color{blue}{\left(\left(0.5 \cdot \frac{1}{{x}^{2}} + 0.0625 \cdot \frac{1}{{x}^{6}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)}\right)\right), x\right) \]

      3. +-commutative 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\color{blue}{\left(0.0625 \cdot \frac{1}{{x}^{6}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)} + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      4. associate-*r/ 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\color{blue}{\frac{0.0625 \cdot 1}{{x}^{6}}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      5. metadata-eval 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{\color{blue}{0.0625}}{{x}^{6}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      6. metadata-eval 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{{x}^{\color{blue}{\left(2 \cdot 3\right)}}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      7. pow-sqr 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\color{blue}{{x}^{3} \cdot {x}^{3}}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      8. cube-unmult 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot {x}^{3}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      9. cube-unmult 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      10. unpow2 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + 0.5 \cdot \frac{1}{\color{blue}{x \cdot x}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      11. associate-*r/ 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \color{blue}{\frac{0.5 \cdot 1}{x \cdot x}}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      12. metadata-eval 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\color{blue}{0.5}}{x \cdot x}\right) + \left(-\frac{0.125}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      13. metadata-eval 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \left(-\frac{0.125}{{x}^{\color{blue}{\left(3 + 1\right)}}}\right)\right)\right)\right), x\right) \]

      14. pow-plus 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \left(-\frac{0.125}{\color{blue}{{x}^{3} \cdot x}}\right)\right)\right)\right), x\right) \]

      15. unpow3 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \left(-\frac{0.125}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot x}\right)\right)\right)\right), x\right) \]

      16. associate-*r* 98.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \left(-\frac{0.125}{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}\right)\right)\right)\right), x\right) \]

   7. Simplified 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\left(\frac{0.125}{x \cdot x} - 0.5\right) - \frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\left(\frac{0.0625}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{0.5}{x \cdot x}\right) + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\left(\frac{0.125}{x \cdot x} - 0.5\right) - \frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\left(\frac{0.5}{x \cdot x} + \frac{x}{x}\right) + \left(1 - \frac{0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign
    (log (/ (- (- (/ 0.125 (* x x)) 0.5) (/ 0.0625 (* x (* x (* x x))))) x))
    x)
   (if (<= x 1.0)
     (copysign
      (+
       x
       (*
        x
        (*
         (* x x)
         (+
          (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
          -0.16666666666666666))))
      x)
     (copysign
      (log
       (*
        x
        (+ (+ (/ 0.5 (* x x)) (/ x x)) (- 1.0 (/ 0.125 (* (* x x) (* x x)))))))
      x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((((0.125f / (x * x)) - 0.5f) - (0.0625f / (x * (x * (x * x))))) / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x * (((0.5f / (x * x)) + (x / x)) + (1.0f - (0.125f / ((x * x) * (x * x))))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(Float32(Float32(Float32(0.125) / Float32(x * x)) - Float32(0.5)) - Float32(Float32(0.0625) / Float32(x * Float32(x * Float32(x * x))))) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(Float32(Float32(0.5) / Float32(x * x)) + Float32(x / x)) + Float32(Float32(1.0) - Float32(Float32(0.125) / Float32(Float32(x * x) * Float32(x * 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(0.125) / (x * x)) - single(0.5)) - (single(0.0625) / (x * (x * (x * x))))) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x * (((single(0.5) / (x * x)) + (x / x)) + (single(1.0) - (single(0.125) / ((x * x) * (x * x))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in x around -inf 99.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]

      2. neg-sub0 99.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]

      3. +-commutative 99.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\color{blue}{\left(\frac{0.0625}{{x}^{4}} + 0.5\right)} - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right), x\right) \]

      4. associate--l+ 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\color{blue}{\frac{0.0625}{{x}^{4}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}}{x}\right), x\right) \]

      5. metadata-eval 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{{x}^{\color{blue}{\left(3 + 1\right)}}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      6. pow-plus 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{{x}^{3} \cdot x}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      7. unpow3 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot x} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      8. associate-*r* 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      9. associate-*l* 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      10. unpow2 99.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - 0.125 \cdot \frac{1}{\color{blue}{x \cdot x}}\right)}{x}\right), x\right) \]

      11. associate-*r/ 99.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - \color{blue}{\frac{0.125 \cdot 1}{x \cdot x}}\right)}{x}\right), x\right) \]

      12. metadata-eval 99.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - \frac{\color{blue}{0.125}}{x \cdot x}\right)}{x}\right), x\right) \]

   7. Simplified 99.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - \frac{0.125}{x \cdot x}\right)}{x}\right)}, x\right) \]

   if -2 < x < 1

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 21.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. unpow2 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 59.9%

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

   3. Taylor expanded in x around inf 98.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\left(1 + \left(\frac{0.5}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right) - 0.125 \cdot \frac{1}{{x}^{4}}\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\color{blue}{\left(\left(\frac{0.5}{{x}^{2}} + \frac{\left|x\right|}{x}\right) + 1\right)} - 0.125 \cdot \frac{1}{{x}^{4}}\right)\right), x\right) \]

      2. associate-*r/ 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\left(\left(\frac{0.5}{{x}^{2}} + \frac{\left|x\right|}{x}\right) + 1\right) - \color{blue}{\frac{0.125 \cdot 1}{{x}^{4}}}\right)\right), x\right) \]

      3. metadata-eval 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\left(\left(\frac{0.5}{{x}^{2}} + \frac{\left|x\right|}{x}\right) + 1\right) - \frac{\color{blue}{0.125}}{{x}^{4}}\right)\right), x\right) \]

      4. associate--l+ 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \color{blue}{\left(\left(\frac{0.5}{{x}^{2}} + \frac{\left|x\right|}{x}\right) + \left(1 - \frac{0.125}{{x}^{4}}\right)\right)}\right), x\right) \]

      5. +-commutative 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\color{blue}{\left(\frac{\left|x\right|}{x} + \frac{0.5}{{x}^{2}}\right)} + \left(1 - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      6. rem-square-sqrt 98.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{x} + \frac{0.5}{{x}^{2}}\right) + \left(1 - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      7. fabs-sqr 98.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x} + \frac{0.5}{{x}^{2}}\right) + \left(1 - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      8. rem-square-sqrt 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\left(\frac{\color{blue}{x}}{x} + \frac{0.5}{{x}^{2}}\right) + \left(1 - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      9. unpow2 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(\left(\frac{x}{x} + \frac{0.5}{\color{blue}{x \cdot x}}\right) + \left(1 - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      10. metadata-eval 98.2%

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

      11. pow-plus 98.2%

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

      12. unpow3 98.2%

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

      13. associate-*r* 98.2%

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

   5. Simplified 98.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\left(\frac{x}{x} + \frac{0.5}{x \cdot x}\right) + \left(1 - \frac{0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\left(\frac{0.125}{x \cdot x} - 0.5\right) - \frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\left(\frac{0.5}{x \cdot x} + \frac{x}{x}\right) + \left(1 - \frac{0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\left(\frac{0.125}{x \cdot x} - 0.5\right) - \frac{0.0625}{t\_0}}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \frac{-0.125}{t\_0}\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -2.0)
     (copysign (log (/ (- (- (/ 0.125 (* x x)) 0.5) (/ 0.0625 t_0)) x)) x)
     (if (<= x 1.0)
       (copysign
        (+
         x
         (*
          x
          (*
           (* x x)
           (+
            (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
            -0.16666666666666666))))
        x)
       (copysign (log (* x (+ 2.0 (+ (/ 0.5 (* x x)) (/ -0.125 t_0))))) x)))))

C

float code(float x) {
	float t_0 = x * (x * (x * x));
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(((((0.125f / (x * x)) - 0.5f) - (0.0625f / t_0)) / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x * (2.0f + ((0.5f / (x * x)) + (-0.125f / t_0))))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(x * Float32(x * Float32(x * x)))
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(Float32(Float32(Float32(0.125) / Float32(x * x)) - Float32(0.5)) - Float32(Float32(0.0625) / t_0)) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(2.0) + Float32(Float32(Float32(0.5) / Float32(x * x)) + Float32(Float32(-0.125) / t_0))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * (x * x));
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log(((((single(0.125) / (x * x)) - single(0.5)) - (single(0.0625) / t_0)) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x * (single(2.0) + ((single(0.5) / (x * x)) + (single(-0.125) / t_0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in x around -inf 99.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]

      2. neg-sub0 99.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \frac{\left(0.5 + \frac{0.0625}{{x}^{4}}\right) - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]

      3. +-commutative 99.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\color{blue}{\left(\frac{0.0625}{{x}^{4}} + 0.5\right)} - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right), x\right) \]

      4. associate--l+ 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\color{blue}{\frac{0.0625}{{x}^{4}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}}{x}\right), x\right) \]

      5. metadata-eval 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{{x}^{\color{blue}{\left(3 + 1\right)}}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      6. pow-plus 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{{x}^{3} \cdot x}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      7. unpow3 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot x} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      8. associate-*r* 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      9. associate-*l* 99.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} + \left(0.5 - 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right), x\right) \]

      10. unpow2 99.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - 0.125 \cdot \frac{1}{\color{blue}{x \cdot x}}\right)}{x}\right), x\right) \]

      11. associate-*r/ 99.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - \color{blue}{\frac{0.125 \cdot 1}{x \cdot x}}\right)}{x}\right), x\right) \]

      12. metadata-eval 99.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - \frac{\color{blue}{0.125}}{x \cdot x}\right)}{x}\right), x\right) \]

   7. Simplified 99.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \frac{\frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(0.5 - \frac{0.125}{x \cdot x}\right)}{x}\right)}, x\right) \]

   if -2 < x < 1

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 21.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. unpow2 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

      2. +-commutative 59.9%

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

      3. hypot-1-def 98.5%

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

      4. add-sqr-sqrt 98.5%

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

      5. fabs-sqr 98.5%

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

      6. add-sqr-sqrt 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in x around inf 98.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\left(2 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{0.125}{{x}^{4}}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. associate--l+ 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \color{blue}{\left(2 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{0.125}{{x}^{4}}\right)\right)}\right), x\right) \]

      2. unpow2 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(0.5 \cdot \frac{1}{\color{blue}{x \cdot x}} - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      3. associate-*r/ 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\color{blue}{\frac{0.5 \cdot 1}{x \cdot x}} - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      4. metadata-eval 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{\color{blue}{0.5}}{x \cdot x} - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      5. sub-neg 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \color{blue}{\left(\frac{0.5}{x \cdot x} + \left(-\frac{0.125}{{x}^{4}}\right)\right)}\right)\right), x\right) \]

      6. metadata-eval 98.2%

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

      7. pow-plus 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \left(-\frac{0.125}{\color{blue}{{x}^{3} \cdot x}}\right)\right)\right)\right), x\right) \]

      8. unpow3 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \left(-\frac{0.125}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot x}\right)\right)\right)\right), x\right) \]

      9. associate-*r* 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \left(-\frac{0.125}{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}\right)\right)\right)\right), x\right) \]

      10. distribute-neg-frac 98.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \color{blue}{\frac{-0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}\right)\right)\right), x\right) \]

      11. metadata-eval 98.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \frac{\color{blue}{-0.125}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right)\right), x\right) \]

      12. associate-*l* 98.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \frac{-0.125}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}\right)\right)\right), x\right) \]

   7. Simplified 98.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\left(\frac{0.125}{x \cdot x} - 0.5\right) - \frac{0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\frac{0.125}{x \cdot x} - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ (- (/ 0.125 (* x x)) 0.5) x)) x)
   (if (<= x 1.0)
     (copysign
      (+
       x
       (*
        x
        (*
         (* x x)
         (+
          (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
          -0.16666666666666666))))
      x)
     (copysign
      (log (* x (+ 2.0 (+ (/ 0.5 (* x x)) (/ -0.125 (* x (* x (* x x))))))))
      x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((((0.125f / (x * x)) - 0.5f) / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x * (2.0f + ((0.5f / (x * x)) + (-0.125f / (x * (x * (x * x)))))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(Float32(Float32(0.125) / Float32(x * x)) - Float32(0.5)) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(2.0) + Float32(Float32(Float32(0.5) / Float32(x * x)) + Float32(Float32(-0.125) / Float32(x * Float32(x * Float32(x * 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(0.125) / (x * x)) - single(0.5)) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x * (single(2.0) + ((single(0.5) / (x * x)) + (single(-0.125) / (x * (x * (x * x)))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in x around -inf 99.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]

      2. distribute-neg-frac2 99.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{-x}\right)}, x\right) \]

      3. unpow2 99.0%

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

      4. associate-*r/ 99.0%

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

      5. metadata-eval 99.0%

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

      6. neg-mul-1 99.0%

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

      7. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   if -2 < x < 1

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 21.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. unpow2 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

      2. +-commutative 59.9%

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

      3. hypot-1-def 98.5%

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

      4. add-sqr-sqrt 98.5%

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

      5. fabs-sqr 98.5%

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

      6. add-sqr-sqrt 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in x around inf 98.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\left(2 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{0.125}{{x}^{4}}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. associate--l+ 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \color{blue}{\left(2 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{0.125}{{x}^{4}}\right)\right)}\right), x\right) \]

      2. unpow2 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(0.5 \cdot \frac{1}{\color{blue}{x \cdot x}} - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      3. associate-*r/ 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\color{blue}{\frac{0.5 \cdot 1}{x \cdot x}} - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      4. metadata-eval 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{\color{blue}{0.5}}{x \cdot x} - \frac{0.125}{{x}^{4}}\right)\right)\right), x\right) \]

      5. sub-neg 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \color{blue}{\left(\frac{0.5}{x \cdot x} + \left(-\frac{0.125}{{x}^{4}}\right)\right)}\right)\right), x\right) \]

      6. metadata-eval 98.2%

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

      7. pow-plus 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \left(-\frac{0.125}{\color{blue}{{x}^{3} \cdot x}}\right)\right)\right)\right), x\right) \]

      8. unpow3 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \left(-\frac{0.125}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot x}\right)\right)\right)\right), x\right) \]

      9. associate-*r* 98.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \left(-\frac{0.125}{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}\right)\right)\right)\right), x\right) \]

      10. distribute-neg-frac 98.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \color{blue}{\frac{-0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}\right)\right)\right), x\right) \]

      11. metadata-eval 98.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \frac{\color{blue}{-0.125}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right)\right), x\right) \]

      12. associate-*l* 98.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \frac{-0.125}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}\right)\right)\right), x\right) \]

   7. Simplified 98.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\frac{0.125}{x \cdot x} - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\frac{0.125}{x \cdot x} - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \frac{0.5}{x \cdot x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ (- (/ 0.125 (* x x)) 0.5) x)) x)
   (if (<= x 1.0)
     (copysign
      (+
       x
       (*
        x
        (*
         (* x x)
         (+
          (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
          -0.16666666666666666))))
      x)
     (copysign (log (* x (+ 2.0 (/ 0.5 (* x x))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((((0.125f / (x * x)) - 0.5f) / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x * (2.0f + (0.5f / (x * x))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(Float32(Float32(0.125) / Float32(x * x)) - Float32(0.5)) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(2.0) + Float32(Float32(0.5) / Float32(x * 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(0.125) / (x * x)) - single(0.5)) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x * (single(2.0) + (single(0.5) / (x * x))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in x around -inf 99.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-\frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)}, x\right) \]

      2. distribute-neg-frac2 99.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{-x}\right)}, x\right) \]

      3. unpow2 99.0%

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

      4. associate-*r/ 99.0%

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

      5. metadata-eval 99.0%

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

      6. neg-mul-1 99.0%

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

      7. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   if -2 < x < 1

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 21.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. unpow2 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 59.9%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. associate-+r+ 97.6%

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

      2. metadata-eval 97.6%

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

      3. unpow2 97.6%

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

      4. associate-*r/ 97.6%

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

      5. unpow2 97.6%

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

      6. +-commutative 97.6%

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

      7. rem-square-sqrt 97.6%

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

      8. fabs-sqr 97.6%

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

      9. rem-square-sqrt 97.6%

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

      10. unpow2 97.6%

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

      11. associate-*r/ 97.6%

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

      12. metadata-eval 97.6%

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

      13. unpow2 97.6%

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

      14. unpow2 97.6%

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

   5. Simplified 97.6%

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

      1. *-commutative 97.6%

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

      2. *-inverses 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l+ 97.6%

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

      6. metadata-eval 97.6%

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

   7. Applied egg-rr 97.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\frac{0.125}{x \cdot x} - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \frac{0.5}{x \cdot x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(2 + \frac{0.5}{x \cdot x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign
      (+
       x
       (*
        x
        (*
         (* x x)
         (+
          (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
          -0.16666666666666666))))
      x)
     (copysign (log (* x (+ 2.0 (/ 0.5 (* x x))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(logf((x * (2.0f + (0.5f / (x * 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(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(2.0) + Float32(Float32(0.5) / Float32(x * 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(1.0))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	else
		tmp = sign(x) * abs(log((x * (single(2.0) + (single(0.5) / (x * x))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in x around -inf 96.7%

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

   if -2 < x < 1

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 21.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. unpow2 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 59.9%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. associate-+r+ 97.6%

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

      2. metadata-eval 97.6%

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

      3. unpow2 97.6%

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

      4. associate-*r/ 97.6%

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

      5. unpow2 97.6%

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

      6. +-commutative 97.6%

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

      7. rem-square-sqrt 97.6%

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

      8. fabs-sqr 97.6%

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

      9. rem-square-sqrt 97.6%

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

      10. unpow2 97.6%

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

      11. associate-*r/ 97.6%

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

      12. metadata-eval 97.6%

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

      13. unpow2 97.6%

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

      14. unpow2 97.6%

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

   5. Simplified 97.6%

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

      1. *-commutative 97.6%

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

      2. *-inverses 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+l+ 97.6%

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

      6. metadata-eval 97.6%

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

   7. Applied egg-rr 97.6%

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

4. Final simplification 98.5%

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

Alternative 11: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign
      (+
       x
       (*
        x
        (*
         (* x x)
         (+
          (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
          -0.16666666666666666))))
      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 <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), 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(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), 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(1.0))
		tmp = sign(x) * abs((x + (x * ((x * x) * (((x * x) * (((x * x) * single(-0.044642857142857144)) + single(0.075))) + single(-0.16666666666666666))))));
	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 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in x around -inf 96.7%

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

   if -2 < x < 1

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 21.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. unpow2 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 59.9%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. associate-+r+ 97.6%

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

      2. metadata-eval 97.6%

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

      3. unpow2 97.6%

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

      4. associate-*r/ 97.6%

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

      5. unpow2 97.6%

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

      6. +-commutative 97.6%

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

      7. rem-square-sqrt 97.6%

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

      8. fabs-sqr 97.6%

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

      9. rem-square-sqrt 97.6%

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

      10. unpow2 97.6%

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

      11. associate-*r/ 97.6%

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

      12. metadata-eval 97.6%

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

      13. unpow2 97.6%

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

      14. unpow2 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in x around 0 97.6%

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

4. Final simplification 98.4%

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

Alternative 12: 85.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign
      (+
       x
       (*
        x
        (*
         (* x x)
         (+
          (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
          -0.16666666666666666))))
      x)
     (copysign (log1p x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(log1pf(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(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      2. +-commutative 55.5%

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

      3. hypot-1-def 99.9%

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

      4. add-sqr-sqrt -0.0%

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

      5. fabs-sqr -0.0%

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

      6. add-sqr-sqrt 15.1%

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

   4. Applied egg-rr 15.1%

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

   5. Taylor expanded in x around -inf 96.7%

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

   if -2 < x < 1

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 21.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. unpow2 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 59.9%

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

   3. Taylor expanded in x around 0 43.7%

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

      1. log1p-define 43.7%

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

      2. rem-square-sqrt 43.7%

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

      3. fabs-sqr 43.7%

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

      4. rem-square-sqrt 43.7%

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

   5. Simplified 43.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (- x)) x)
   (if (<= x 1.0)
     (copysign
      (+
       x
       (*
        x
        (*
         (* x x)
         (+
          (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
          -0.16666666666666666))))
      x)
     (copysign (log1p x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 55.5%

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

   3. Taylor expanded in x around -inf 43.5%

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

   if -2 < x < 1

   1. Initial program 21.8%

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

   3. Taylor expanded in x around 0 21.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 10.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 21.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 21.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 21.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. unpow2 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1 < x

   1. Initial program 59.9%

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

   3. Taylor expanded in x around 0 43.7%

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

      1. log1p-define 43.7%

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

      2. rem-square-sqrt 43.7%

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

      3. fabs-sqr 43.7%

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

      4. rem-square-sqrt 43.7%

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

   5. Simplified 43.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 1.0)
   (copysign
    (+
     x
     (*
      x
      (*
       (* x x)
       (+
        (* (* x x) (+ (* (* x x) -0.044642857142857144) 0.075))
        -0.16666666666666666))))
    x)
   (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 1.0f) {
		tmp = copysignf((x + (x * ((x * x) * (((x * x) * (((x * x) * -0.044642857142857144f) + 0.075f)) + -0.16666666666666666f)))), x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.0))
		tmp = copysign(Float32(x + Float32(x * Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(x * x) * Float32(-0.044642857142857144)) + Float32(0.075))) + Float32(-0.16666666666666666))))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 32.1%

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

   3. Taylor expanded in x around 0 18.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. rem-square-sqrt 7.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      2. fabs-sqr 7.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      3. rem-square-sqrt 18.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x} + {x}^{2} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      4. unpow2 18.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right), x\right) \]

      5. associate-*l* 18.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{x \cdot \left(x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right)\right), x\right) \]

      6. unpow2 18.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)\right)\right), x\right) \]

      7. *-commutative 18.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} - 0.125\right) \cdot \left(x \cdot x\right)}\right)\right)\right)\right), x\right) \]

      8. sub-neg 18.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(0.0625 \cdot {x}^{2} + \left(-0.125\right)\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      9. metadata-eval 18.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + \color{blue}{-0.125}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      10. +-commutative 18.8%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(-0.125 + 0.0625 \cdot {x}^{2}\right)} \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      11. unpow2 18.8%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 18.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + x \cdot \left(x \cdot \left(0.5 + \left(-0.125 + 0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0 71.4%

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

      1. unpow2 71.4%

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

      2. *-commutative 71.4%

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

      3. sub-neg 71.4%

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

      4. unpow2 71.4%

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

      5. *-commutative 71.4%

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

      6. unpow2 71.4%

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

      7. metadata-eval 71.4%

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

   8. Simplified 71.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \left(\left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666\right) \cdot \left(x \cdot x\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 71.4%

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

      2. *-un-lft-identity 71.4%

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

      3. *-commutative 71.4%

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

      4. *-commutative 71.4%

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

      5. +-commutative 71.4%

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

      6. *-commutative 71.4%

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

   10. Applied egg-rr 71.4%

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

   if 1 < x

   1. Initial program 59.9%

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

   3. Taylor expanded in x around 0 43.7%

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

      1. log1p-define 43.7%

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

      2. rem-square-sqrt 43.7%

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

      3. fabs-sqr 43.7%

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

      4. rem-square-sqrt 43.7%

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

   5. Simplified 43.7%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144 + 0.075\right) + -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.2% 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 39.0%

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

3. Taylor expanded in x around 0 30.4%

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

   1. rem-square-sqrt 15.6%

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

   2. fabs-sqr 15.6%

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

   3. rem-square-sqrt 20.4%

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

5. Simplified 20.4%

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

6. Taylor expanded in x around 0 56.5%

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

Developer target: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

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

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