Rust f32::asinh

Percentage Accurate: 37.6% → 98.5%

Time: 16.8s

Alternatives: 19

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: 37.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left(x \cdot x\right) \cdot 0.001388888888888889\\ t_2 := \frac{0.125}{x \cdot x}\\ t_3 := \left|x\right| + 1\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5 + t\_2}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(\frac{1}{t\_3} + \frac{1}{{t\_3}^{2}}\right) \cdot \left(-0.125 + t\_1 \cdot 45\right) + \frac{t\_1 \cdot 30}{{t\_3}^{3}}\right)\right) + \frac{0.5}{t\_3}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \frac{\left|x\right| + \frac{0.5 - t\_2}{x}}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (* (* x x) 0.001388888888888889))
        (t_2 (/ 0.125 (* x x)))
        (t_3 (+ (fabs x) 1.0)))
   (if (<= t_0 -2.0)
     (copysign (log (- (+ (fabs x) (/ (+ -0.5 t_2) x)) x)) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (+
         (log1p (fabs x))
         (*
          (* x x)
          (+
           (*
            x
            (*
             x
             (+
              (* (+ (/ 1.0 t_3) (/ 1.0 (pow t_3 2.0))) (+ -0.125 (* t_1 45.0)))
              (/ (* t_1 30.0) (pow t_3 3.0)))))
           (/ 0.5 t_3))))
        x)
       (copysign
        (log (* x (+ 1.0 (/ (+ (fabs x) (/ (- 0.5 t_2) x)) x))))
        x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = (x * x) * 0.001388888888888889f;
	float t_2 = 0.125f / (x * x);
	float t_3 = fabsf(x) + 1.0f;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = copysignf(logf(((fabsf(x) + ((-0.5f + t_2) / x)) - x)), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf((log1pf(fabsf(x)) + ((x * x) * ((x * (x * ((((1.0f / t_3) + (1.0f / powf(t_3, 2.0f))) * (-0.125f + (t_1 * 45.0f))) + ((t_1 * 30.0f) / powf(t_3, 3.0f))))) + (0.5f / t_3)))), x);
	} else {
		tmp = copysignf(logf((x * (1.0f + ((fabsf(x) + ((0.5f - t_2) / x)) / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = Float32(Float32(x * x) * Float32(0.001388888888888889))
	t_2 = Float32(Float32(0.125) / Float32(x * x))
	t_3 = Float32(abs(x) + Float32(1.0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) + Float32(Float32(Float32(-0.5) + t_2) / x)) - x)), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(log1p(abs(x)) + Float32(Float32(x * x) * Float32(Float32(x * Float32(x * Float32(Float32(Float32(Float32(Float32(1.0) / t_3) + Float32(Float32(1.0) / (t_3 ^ Float32(2.0)))) * Float32(Float32(-0.125) + Float32(t_1 * Float32(45.0)))) + Float32(Float32(t_1 * Float32(30.0)) / (t_3 ^ Float32(3.0)))))) + Float32(Float32(0.5) / t_3)))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(1.0) + Float32(Float32(abs(x) + Float32(Float32(Float32(0.5) - t_2) / x)) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 56.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

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

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

   5. Simplified 99.9%

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

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

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{24} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) + \frac{1}{720} \cdot \left({x}^{2} \cdot \left(45 \cdot \frac{1}{1 + \left|x\right|} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}}\right)\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)}, x\right) \]

   5. Simplified 99.8%

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

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

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around -inf

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

   8. Simplified 99.3%

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

4. Final simplification 99.7%

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

Alternative 2: 98.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (/ 0.125 (* x x))))
   (if (<= t_0 -2.0)
     (copysign (log (- (+ (fabs x) (/ (+ -0.5 t_1) x)) x)) x)
     (if (<= t_0 0.20000000298023224)
       (copysign (log1p (+ (fabs x) (* (* x x) (+ 0.5 (* x (* x -0.125)))))) x)
       (copysign
        (log (* x (+ 1.0 (/ (+ (fabs x) (/ (- 0.5 t_1) x)) x))))
        x)))))

C

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

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = Float32(Float32(0.125) / Float32(x * x))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(abs(x) + Float32(Float32(Float32(-0.5) + t_1) / x)) - x)), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(x * Float32(x * Float32(-0.125))))))), x);
	else
		tmp = copysign(log(Float32(x * Float32(Float32(1.0) + Float32(Float32(abs(x) + Float32(Float32(Float32(0.5) - t_1) / x)) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 56.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

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

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

   5. Simplified 99.9%

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

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

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 20.6%

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

   5. Simplified 20.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-log1p.f32 N/A

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

      10. associate-*l* N/A

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

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

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

      12. *-lowering-*.f32 99.6%

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

   7. Applied egg-rr 99.6%

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

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

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around -inf

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

   8. Simplified 99.3%

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

Alternative 3: 98.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{x \cdot x}\\ \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5 + t\_0}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot -0.125\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5 - t\_0}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* x x))))
   (if (<= x -10.0)
     (copysign (log (- (+ (fabs x) (/ (+ -0.5 t_0) x)) x)) x)
     (if (<= x 0.20000000298023224)
       (copysign (log1p (+ (fabs x) (* (* x x) (+ 0.5 (* x (* x -0.125)))))) x)
       (copysign (log (+ (fabs x) (+ x (/ (- 0.5 t_0) x)))) x)))))

C

float code(float x) {
	float t_0 = 0.125f / (x * x);
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(((fabsf(x) + ((-0.5f + t_0) / x)) - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(log1pf((fabsf(x) + ((x * x) * (0.5f + (x * (x * -0.125f)))))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + ((0.5f - t_0) / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(Float32(0.125) / Float32(x * x))
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(abs(x) + Float32(Float32(Float32(-0.5) + t_0) / x)) - x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(Float32(0.5) + Float32(x * Float32(x * Float32(-0.125))))))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(Float32(0.5) - t_0) / x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

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

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

   5. Simplified 99.9%

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

   if -10 < x < 0.200000003

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 20.6%

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

   5. Simplified 20.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-log1p.f32 N/A

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

      10. associate-*l* N/A

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

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

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

      12. *-lowering-*.f32 99.6%

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

   7. Applied egg-rr 99.6%

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

   if 0.200000003 < x

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot -0.125\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5 - \frac{0.125}{x \cdot x}}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{x \cdot x}\\ \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5 + t\_0}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5 - t\_0}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* x x))))
   (if (<= x -10.0)
     (copysign (log (- (+ (fabs x) (/ (+ -0.5 t_0) x)) x)) x)
     (if (<= x 0.20000000298023224)
       (copysign
        (*
         x
         (+
          1.0
          (*
           x
           (+
            (/ 0.5 (+ (fabs x) 1.0))
            (+ -0.5 (* x (+ 0.3333333333333333 (* x -0.25))))))))
        x)
       (copysign (log (+ (fabs x) (+ x (/ (- 0.5 t_0) x)))) x)))))

C

float code(float x) {
	float t_0 = 0.125f / (x * x);
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(((fabsf(x) + ((-0.5f + t_0) / x)) - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (x * ((0.5f / (fabsf(x) + 1.0f)) + (-0.5f + (x * (0.3333333333333333f + (x * -0.25f)))))))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + ((0.5f - t_0) / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(Float32(0.125) / Float32(x * x))
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(abs(x) + Float32(Float32(Float32(-0.5) + t_0) / x)) - x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(x * Float32(Float32(Float32(0.5) / Float32(abs(x) + Float32(1.0))) + Float32(Float32(-0.5) + Float32(x * Float32(Float32(0.3333333333333333) + Float32(x * Float32(-0.25))))))))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(Float32(0.5) - t_0) / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = single(0.125) / (x * x);
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log(((abs(x) + ((single(-0.5) + t_0) / x)) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x * (single(1.0) + (x * ((single(0.5) / (abs(x) + single(1.0))) + (single(-0.5) + (x * (single(0.3333333333333333) + (x * single(-0.25))))))))));
	else
		tmp = sign(x) * abs(log((abs(x) + (x + ((single(0.5) - t_0) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

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

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

   5. Simplified 99.9%

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

   if -10 < x < 0.200000003

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

      7. accelerator-lowering-log1p.f32 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f32 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. +-commutative N/A

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

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

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

   5. Simplified 98.6%

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

   7. Applied egg-rr 15.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2} \cdot 1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2}}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left(1 + \left|x\right|\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \left(\left|x\right|\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(\frac{-1}{2} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 98.2%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{0.5}{1 + \left|x\right|} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right)}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 99.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5 - \frac{0.125}{x \cdot x}}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5 - \frac{0.125}{x \cdot x}}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
   (if (<= x 0.20000000298023224)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (+
          (/ 0.5 (+ (fabs x) 1.0))
          (+ -0.5 (* x (+ 0.3333333333333333 (* x -0.25))))))))
      x)
     (copysign (log (+ (fabs x) (+ x (/ (- 0.5 (/ 0.125 (* x x))) x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (x * ((0.5f / (fabsf(x) + 1.0f)) + (-0.5f + (x * (0.3333333333333333f + (x * -0.25f)))))))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x + ((0.5f - (0.125f / (x * x))) / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(x * Float32(Float32(Float32(0.5) / Float32(abs(x) + Float32(1.0))) + Float32(Float32(-0.5) + Float32(x * Float32(Float32(0.3333333333333333) + Float32(x * Float32(-0.25))))))))), x);
	else
		tmp = copysign(log(Float32(abs(x) + Float32(x + Float32(Float32(Float32(0.5) - Float32(Float32(0.125) / Float32(x * x))) / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log(((abs(x) - x) + (single(-0.5) / x))));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x * (single(1.0) + (x * ((single(0.5) / (abs(x) + single(1.0))) + (single(-0.5) + (x * (single(0.3333333333333333) + (x * single(-0.25))))))))));
	else
		tmp = sign(x) * abs(log((abs(x) + (x + ((single(0.5) - (single(0.125) / (x * x))) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.8%

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

   if -10 < x < 0.200000003

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

      7. accelerator-lowering-log1p.f32 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f32 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. +-commutative N/A

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

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

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

   5. Simplified 98.6%

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

   7. Applied egg-rr 15.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2} \cdot 1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2}}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left(1 + \left|x\right|\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \left(\left|x\right|\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(\frac{-1}{2} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 98.2%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{0.5}{1 + \left|x\right|} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right)}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf

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

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

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

   8. Simplified 99.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5 - \frac{0.125}{x \cdot x}}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + \left|x\right|\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
   (if (<= x 0.20000000298023224)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (+
          (/ 0.5 (+ (fabs x) 1.0))
          (+ -0.5 (* x (+ 0.3333333333333333 (* x -0.25))))))))
      x)
     (copysign (log (+ (+ x (fabs x)) (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(((fabsf(x) - x) + (-0.5f / x))), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (x * ((0.5f / (fabsf(x) + 1.0f)) + (-0.5f + (x * (0.3333333333333333f + (x * -0.25f)))))))), x);
	} else {
		tmp = copysignf(logf(((x + fabsf(x)) + (0.5f / x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(abs(x) - x) + Float32(Float32(-0.5) / x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(x * Float32(Float32(Float32(0.5) / Float32(abs(x) + Float32(1.0))) + Float32(Float32(-0.5) + Float32(x * Float32(Float32(0.3333333333333333) + Float32(x * Float32(-0.25))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(x + abs(x)) + Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log(((abs(x) - x) + (single(-0.5) / x))));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x * (single(1.0) + (x * ((single(0.5) / (abs(x) + single(1.0))) + (single(-0.5) + (x * (single(0.3333333333333333) + (x * single(-0.25))))))))));
	else
		tmp = sign(x) * abs(log(((x + abs(x)) + (single(0.5) / x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.8%

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

   if -10 < x < 0.200000003

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

      7. accelerator-lowering-log1p.f32 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f32 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. +-commutative N/A

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

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

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

   5. Simplified 98.6%

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

   7. Applied egg-rr 15.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2} \cdot 1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2}}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left(1 + \left|x\right|\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \left(\left|x\right|\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(\frac{-1}{2} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 98.2%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{0.5}{1 + \left|x\right|} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right)}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lft-mult-inverse N/A

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

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

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

   5. Simplified 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + \left|x\right|\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + \left|x\right|\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 0.20000000298023224)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (+
          (/ 0.5 (+ (fabs x) 1.0))
          (+ -0.5 (* x (+ 0.3333333333333333 (* x -0.25))))))))
      x)
     (copysign (log (+ (+ x (fabs x)) (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (x * ((0.5f / (fabsf(x) + 1.0f)) + (-0.5f + (x * (0.3333333333333333f + (x * -0.25f)))))))), x);
	} else {
		tmp = copysignf(logf(((x + fabsf(x)) + (0.5f / x))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(x * Float32(Float32(Float32(0.5) / Float32(abs(x) + Float32(1.0))) + Float32(Float32(-0.5) + Float32(x * Float32(Float32(0.3333333333333333) + Float32(x * Float32(-0.25))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(x + abs(x)) + Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((abs(x) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x * (single(1.0) + (x * ((single(0.5) / (abs(x) + single(1.0))) + (single(-0.5) + (x * (single(0.3333333333333333) + (x * single(-0.25))))))))));
	else
		tmp = sign(x) * abs(log(((x + abs(x)) + (single(0.5) / x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. --lowering--.f32 N/A

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

      16. fabs-lowering-fabs.f32 98.8%

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

   5. Simplified 98.8%

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

   if -10 < x < 0.200000003

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

      7. accelerator-lowering-log1p.f32 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f32 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. +-commutative N/A

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

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

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

   5. Simplified 98.6%

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

   7. Applied egg-rr 15.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2} \cdot 1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2}}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left(1 + \left|x\right|\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \left(\left|x\right|\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(\frac{-1}{2} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 98.2%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{0.5}{1 + \left|x\right|} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right)}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lft-mult-inverse N/A

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

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

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

   5. Simplified 98.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + \left|x\right|\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 0.20000000298023224)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (+
          (/ 0.5 (+ (fabs x) 1.0))
          (+ -0.5 (* x (+ 0.3333333333333333 (* x -0.25))))))))
      x)
     (copysign (log (+ (/ 0.5 x) (/ -0.125 (* x (* x x))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (x * ((0.5f / (fabsf(x) + 1.0f)) + (-0.5f + (x * (0.3333333333333333f + (x * -0.25f)))))))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (-0.125f / (x * (x * x))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(x * Float32(Float32(Float32(0.5) / Float32(abs(x) + Float32(1.0))) + Float32(Float32(-0.5) + Float32(x * Float32(Float32(0.3333333333333333) + Float32(x * Float32(-0.25))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(Float32(-0.125) / Float32(x * Float32(x * x))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log((abs(x) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x * (single(1.0) + (x * ((single(0.5) / (abs(x) + single(1.0))) + (single(-0.5) + (x * (single(0.3333333333333333) + (x * single(-0.25))))))))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (single(-0.125) / (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. --lowering--.f32 N/A

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

      16. fabs-lowering-fabs.f32 98.8%

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

   5. Simplified 98.8%

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

   if -10 < x < 0.200000003

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

      7. accelerator-lowering-log1p.f32 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f32 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. +-commutative N/A

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

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

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

   5. Simplified 98.6%

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

   7. Applied egg-rr 15.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2} \cdot 1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2}}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left(1 + \left|x\right|\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \left(\left|x\right|\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(\frac{-1}{2} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 98.2%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{0.5}{1 + \left|x\right|} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right)}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

      20. metadata-eval N/A

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

      21. /-lowering-/.f32 N/A

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

      22. cube-mult N/A

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

      23. unpow2 N/A

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

   8. Simplified 98.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -200000000753664:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125 + x \cdot \left(x \cdot -0.5\right)}{t\_0}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{t\_0}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (* x (* x x))))
   (if (<= x -200000000753664.0)
     (copysign (log (- x)) x)
     (if (<= x -10.0)
       (copysign (log (/ (+ 0.125 (* x (* x -0.5))) t_0)) x)
       (if (<= x 0.20000000298023224)
         (copysign
          (*
           x
           (+
            1.0
            (*
             x
             (+
              (/ 0.5 (+ (fabs x) 1.0))
              (+ -0.5 (* x (+ 0.3333333333333333 (* x -0.25))))))))
          x)
         (copysign (log (+ (/ 0.5 x) (/ -0.125 t_0))) x))))))

C

float code(float x) {
	float t_0 = x * (x * x);
	float tmp;
	if (x <= -200000000753664.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= -10.0f) {
		tmp = copysignf(logf(((0.125f + (x * (x * -0.5f))) / t_0)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (x * ((0.5f / (fabsf(x) + 1.0f)) + (-0.5f + (x * (0.3333333333333333f + (x * -0.25f)))))))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (-0.125f / t_0))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(x * Float32(x * x))
	tmp = Float32(0.0)
	if (x <= Float32(-200000000753664.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(Float32(0.125) + Float32(x * Float32(x * Float32(-0.5)))) / t_0)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(x * Float32(Float32(Float32(0.5) / Float32(abs(x) + Float32(1.0))) + Float32(Float32(-0.5) + Float32(x * Float32(Float32(0.3333333333333333) + Float32(x * Float32(-0.25))))))))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(Float32(-0.125) / t_0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	tmp = single(0.0);
	if (x <= single(-200000000753664.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(-10.0))
		tmp = sign(x) * abs(log(((single(0.125) + (x * (x * single(-0.5)))) / t_0)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x * (single(1.0) + (x * ((single(0.5) / (abs(x) + single(1.0))) + (single(-0.5) + (x * (single(0.3333333333333333) + (x * single(-0.25))))))))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (single(-0.125) / t_0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if x < -2.00000001e14

   1. Initial program 17.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 47.0%

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

   5. Simplified 47.0%

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

   if -2.00000001e14 < x < -10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

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

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. sqr-abs N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

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

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

      22. cube-mult N/A

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

      23. unpow2 N/A

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

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

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

      25. unpow2 N/A

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

   8. Simplified 99.4%

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

   if -10 < x < 0.200000003

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

      7. accelerator-lowering-log1p.f32 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f32 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. +-commutative N/A

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

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

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

   5. Simplified 98.6%

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

   7. Applied egg-rr 15.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \left(1 + x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right) - \frac{1}{2}\right)\right)\right)\right), x\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|} + \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2} \cdot 1}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2}}{1 + \left|x\right|}\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left(1 + \left|x\right|\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \left(\left|x\right|\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \left(\frac{-1}{2} + x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{-1}{4} \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \left(x \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{+.f32}\left(1, \mathsf{fabs.f32}\left(x\right)\right)\right), \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(x, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   10. Simplified 98.2%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{0.5}{1 + \left|x\right|} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right)}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

      20. metadata-eval N/A

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

      21. /-lowering-/.f32 N/A

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

      22. cube-mult N/A

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

      23. unpow2 N/A

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

   8. Simplified 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right)}, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200000000753664:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125 + x \cdot \left(x \cdot -0.5\right)}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot \left(0.3333333333333333 + x \cdot -0.25\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -200000000753664:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125 + x \cdot \left(x \cdot -0.5\right)}{t\_0}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot 0.3333333333333333\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{t\_0}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (* x (* x x))))
   (if (<= x -200000000753664.0)
     (copysign (log (- x)) x)
     (if (<= x -10.0)
       (copysign (log (/ (+ 0.125 (* x (* x -0.5))) t_0)) x)
       (if (<= x 0.20000000298023224)
         (copysign
          (*
           x
           (+
            1.0
            (*
             x
             (+ (/ 0.5 (+ (fabs x) 1.0)) (+ -0.5 (* x 0.3333333333333333))))))
          x)
         (copysign (log (+ (/ 0.5 x) (/ -0.125 t_0))) x))))))

C

float code(float x) {
	float t_0 = x * (x * x);
	float tmp;
	if (x <= -200000000753664.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= -10.0f) {
		tmp = copysignf(logf(((0.125f + (x * (x * -0.5f))) / t_0)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf((x * (1.0f + (x * ((0.5f / (fabsf(x) + 1.0f)) + (-0.5f + (x * 0.3333333333333333f)))))), x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (-0.125f / t_0))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(x * Float32(x * x))
	tmp = Float32(0.0)
	if (x <= Float32(-200000000753664.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(Float32(0.125) + Float32(x * Float32(x * Float32(-0.5)))) / t_0)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32(x * Float32(Float32(Float32(0.5) / Float32(abs(x) + Float32(1.0))) + Float32(Float32(-0.5) + Float32(x * Float32(0.3333333333333333))))))), x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(Float32(-0.125) / t_0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	tmp = single(0.0);
	if (x <= single(-200000000753664.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(-10.0))
		tmp = sign(x) * abs(log(((single(0.125) + (x * (x * single(-0.5)))) / t_0)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs((x * (single(1.0) + (x * ((single(0.5) / (abs(x) + single(1.0))) + (single(-0.5) + (x * single(0.3333333333333333))))))));
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (single(-0.125) / t_0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if x < -2.00000001e14

   1. Initial program 17.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 47.0%

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

   5. Simplified 47.0%

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

   if -2.00000001e14 < x < -10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

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

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. sqr-abs N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

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

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

      22. cube-mult N/A

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

      23. unpow2 N/A

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

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

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

      25. unpow2 N/A

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

   8. Simplified 99.4%

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

   if -10 < x < 0.200000003

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

      7. accelerator-lowering-log1p.f32 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f32 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. +-commutative N/A

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

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

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

   5. Simplified 98.6%

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

   7. Applied egg-rr 15.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\left(\frac{1}{3} \cdot x + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right) - \frac{1}{2}\right)\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f32 98.1%

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

   10. Simplified 98.1%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{0.5}{1 + \left|x\right|} + \left(-0.5 + x \cdot 0.3333333333333333\right)\right)\right)}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f32 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

      20. metadata-eval N/A

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

      21. /-lowering-/.f32 N/A

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

      22. cube-mult N/A

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

      23. unpow2 N/A

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

   8. Simplified 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right)}, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200000000753664:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125 + x \cdot \left(x \cdot -0.5\right)}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(-0.5 + x \cdot 0.3333333333333333\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -200000000753664:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125 + x \cdot \left(x \cdot -0.5\right)}{t\_0}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{t\_0}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (* x (* x x))))
   (if (<= x -200000000753664.0)
     (copysign (log (- x)) x)
     (if (<= x -10.0)
       (copysign (log (/ (+ 0.125 (* x (* x -0.5))) t_0)) x)
       (if (<= x 0.20000000298023224)
         (copysign x x)
         (copysign (log (+ (/ 0.5 x) (/ -0.125 t_0))) x))))))

C

float code(float x) {
	float t_0 = x * (x * x);
	float tmp;
	if (x <= -200000000753664.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= -10.0f) {
		tmp = copysignf(logf(((0.125f + (x * (x * -0.5f))) / t_0)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (-0.125f / t_0))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(x * Float32(x * x))
	tmp = Float32(0.0)
	if (x <= Float32(-200000000753664.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(-10.0))
		tmp = copysign(log(Float32(Float32(Float32(0.125) + Float32(x * Float32(x * Float32(-0.5)))) / t_0)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(Float32(-0.125) / t_0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	tmp = single(0.0);
	if (x <= single(-200000000753664.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(-10.0))
		tmp = sign(x) * abs(log(((single(0.125) + (x * (x * single(-0.5)))) / t_0)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (single(-0.125) / t_0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if x < -2.00000001e14

   1. Initial program 17.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 47.0%

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

   5. Simplified 47.0%

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

   if -2.00000001e14 < x < -10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

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

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. sqr-abs N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

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

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

      22. cube-mult N/A

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

      23. unpow2 N/A

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

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

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

      25. unpow2 N/A

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

   8. Simplified 99.4%

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

   if -10 < x < 0.200000003

   1. Initial program 21.0%

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

   3. Taylor expanded in x around 0

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

      1. accelerator-lowering-log1p.f32 N/A

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

      2. fabs-lowering-fabs.f32 94.6%

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

   5. Simplified 94.6%

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

   7. Applied egg-rr 12.2%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 98.0%

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

   if 0.200000003 < x

   1. Initial program 61.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\frac{{x}^{2} \cdot \frac{1}{2}}{{x}^{2} \cdot x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\frac{{x}^{2}}{{x}^{2}} \cdot \frac{\frac{1}{2}}{x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 \cdot \frac{\frac{1}{2}}{x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 \cdot \frac{\frac{1}{2} \cdot 1}{x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 \cdot \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \left(\frac{\frac{1}{2}}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      16. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      19. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{3}}\right)\right)\right), x\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\frac{\frac{-1}{8}}{{x}^{3}}\right)\right)\right), x\right) \]

      21. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left({x}^{3}\right)\right)\right)\right), x\right) \]

      22. cube-mult N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left(x \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

   8. Simplified 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right)}, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200000000753664:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125 + x \cdot \left(x \cdot -0.5\right)}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (- (log (- 1.0 x))) x)
   (if (<= x 0.20000000298023224)
     (copysign x x)
     (copysign (log (+ (/ 0.5 x) (/ -0.125 (* x (* x x))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(-logf((1.0f - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(((0.5f / x) + (-0.125f / (x * (x * x))))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) - x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) / x) + Float32(Float32(-0.125) / Float32(x * Float32(x * x))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(-log((single(1.0) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(((single(0.5) / x) + (single(-0.125) / (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 0.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + -1 \cdot x\right)}\right)\right), x\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), x\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 - x\right)\right)\right), x\right) \]

      3. --lowering--.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, x\right)\right)\right), x\right) \]

   10. Simplified 43.9%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(1 - x\right)}, x\right) \]

   if -10 < x < 0.200000003

   1. Initial program 21.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 94.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 94.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 12.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 98.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right) - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)}\right), x\right) \]

   5. Simplified 99.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \frac{0.5}{x}\right) + x \cdot \left(1 + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}, x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{{x}^{3}} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{{x}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{{x}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{{x}^{3}} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{{x}^{3}}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\frac{{x}^{2} \cdot \frac{1}{2}}{{x}^{3}}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      7. unpow3 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\frac{{x}^{2} \cdot \frac{1}{2}}{\left(x \cdot x\right) \cdot x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\frac{{x}^{2} \cdot \frac{1}{2}}{{x}^{2} \cdot x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      9. times-frac N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\frac{{x}^{2}}{{x}^{2}} \cdot \frac{\frac{1}{2}}{x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 \cdot \frac{\frac{1}{2}}{x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 \cdot \frac{\frac{1}{2} \cdot 1}{x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(1 \cdot \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \left(\frac{\frac{1}{2}}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      16. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      19. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{3}}\right)\right)\right), x\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \left(\frac{\frac{-1}{8}}{{x}^{3}}\right)\right)\right), x\right) \]

      21. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left({x}^{3}\right)\right)\right)\right), x\right) \]

      22. cube-mult N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), \mathsf{/.f32}\left(\frac{-1}{8}, \left(x \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

   8. Simplified 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5 - \frac{0.125}{x \cdot x}}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (- (log (- 1.0 x))) x)
   (if (<= x 0.20000000298023224)
     (copysign x x)
     (copysign (log (/ (- 0.5 (/ 0.125 (* x x))) x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(-logf((1.0f - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(((0.5f - (0.125f / (x * x))) / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) - x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(Float32(0.5) - Float32(Float32(0.125) / Float32(x * x))) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(-log((single(1.0) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(((single(0.5) - (single(0.125) / (x * x))) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 0.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + -1 \cdot x\right)}\right)\right), x\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), x\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 - x\right)\right)\right), x\right) \]

      3. --lowering--.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, x\right)\right)\right), x\right) \]

   10. Simplified 43.9%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(1 - x\right)}, x\right) \]

   if -10 < x < 0.200000003

   1. Initial program 21.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 94.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 94.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 12.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 98.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right) - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)}\right), x\right) \]

   5. Simplified 99.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \frac{0.5}{x}\right) + x \cdot \left(1 + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left|x\right| + \left(\frac{\frac{1}{2}}{x} + x \cdot \left(1 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)\right), x\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left|x\right| \cdot \left|x\right| - \left(\frac{\frac{1}{2}}{x} + x \cdot \left(1 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \left(\frac{\frac{1}{2}}{x} + x \cdot \left(1 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left|x\right| - \left(\frac{\frac{1}{2}}{x} + x \cdot \left(1 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}\right)\right), x\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right| \cdot \left|x\right| - \left(\frac{\frac{1}{2}}{x} + x \cdot \left(1 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \left(\frac{\frac{1}{2}}{x} + x \cdot \left(1 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right), \left(\left|x\right| - \left(\frac{\frac{1}{2}}{x} + x \cdot \left(1 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 4.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{x \cdot x - \left(\frac{0.5}{x} + x \cdot \left(1 + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \left(\frac{0.5}{x} + x \cdot \left(1 + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}{\left|x\right| - \left(\frac{0.5}{x} + x \cdot \left(1 + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   9. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{{x}^{3}} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{{x}^{2} \cdot \frac{1}{2}}{{x}^{3}} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      3. unpow3 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{{x}^{2} \cdot \frac{1}{2}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{{x}^{2} \cdot \frac{1}{2}}{{x}^{2} \cdot x} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{{x}^{2}}{{x}^{2}} \cdot \frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      6. *-inverses N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(1 \cdot \frac{\frac{1}{2} \cdot 1}{x} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(1 \cdot \left(\frac{1}{2} \cdot \frac{1}{x}\right) - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(1 \cdot \frac{1}{2}\right) \cdot \frac{1}{x} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      11. unpow3 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right), x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{8}}{{x}^{2} \cdot x}\right)\right), x\right) \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{8}}{{x}^{2}}}{x}\right)\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{8} \cdot 1}{{x}^{2}}}{x}\right)\right), x\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right), x\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2} \cdot 1}{x} - \frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right), x\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right), x\right) \]

      18. div-sub N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right), x\right) \]

      19. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right), x\right)\right), x\right) \]

   10. Simplified 98.5%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5 - \frac{0.125}{x \cdot x}}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 83.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{x}{0.5}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (- (log (- 1.0 x))) x)
   (if (<= x 0.20000000298023224)
     (copysign x x)
     (copysign (- (log (/ x 0.5))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(-logf((1.0f - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(-logf((x / 0.5f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) - x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(-log(Float32(x / Float32(0.5)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(-log((single(1.0) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(-log((x / single(0.5))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 0.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + -1 \cdot x\right)}\right)\right), x\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), x\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 - x\right)\right)\right), x\right) \]

      3. --lowering--.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, x\right)\right)\right), x\right) \]

   10. Simplified 43.9%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(1 - x\right)}, x\right) \]

   if -10 < x < 0.200000003

   1. Initial program 21.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 94.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 94.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 12.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 98.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \left(\frac{\left|x\right|}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(\left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right), x\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + x \cdot \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right), x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot x\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \left(\frac{\frac{1}{x}}{x} \cdot x\right)\right)\right), x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \frac{\frac{1}{x} \cdot x}{x}\right)\right), x\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right) \]

   5. Simplified 98.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(x + \left|x\right|\right) + \frac{0.5}{x}\right)}, x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 97.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), x\right) \]

   8. Simplified 97.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5}{x}\right)}, x\right) \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log \left(\frac{1}{\frac{x}{\frac{1}{2}}}\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{x}{\frac{1}{2}}\right)\right)\right), x\right) \]

      3. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\log \left(\frac{x}{\frac{1}{2}}\right)\right), x\right) \]

      4. log-lowering-log.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{x}{\frac{1}{2}}\right)\right)\right), x\right) \]

      5. /-lowering-/.f32 97.7%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(x, \frac{1}{2}\right)\right)\right), x\right) \]

   10. Applied egg-rr 97.7%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x}{0.5}\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 83.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (- (log (- 1.0 x))) x)
   (if (<= x 0.20000000298023224)
     (copysign x x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(-logf((1.0f - x)), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) - x))), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(-log((single(1.0) - x)));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 0.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + -1 \cdot x\right)}\right)\right), x\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), x\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(1 - x\right)\right)\right), x\right) \]

      3. --lowering--.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, x\right)\right)\right), x\right) \]

   10. Simplified 43.9%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(1 - x\right)}, x\right) \]

   if -10 < x < 0.200000003

   1. Initial program 21.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 94.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 94.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 12.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 98.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \left(\frac{\left|x\right|}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(\left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right), x\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + x \cdot \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right), x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot x\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \left(\frac{\frac{1}{x}}{x} \cdot x\right)\right)\right), x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \frac{\frac{1}{x} \cdot x}{x}\right)\right), x\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right) \]

   5. Simplified 98.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(x + \left|x\right|\right) + \frac{0.5}{x}\right)}, x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 97.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), x\right) \]

   8. Simplified 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. Add Preprocessing

Alternative 16: 83.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (- x)) x)
   (if (<= x 0.20000000298023224)
     (copysign x x)
     (copysign (log (/ 0.5 x)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{neg.f32}\left(x\right)\right), x\right) \]

   5. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   if -10 < x < 0.200000003

   1. Initial program 21.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 94.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 94.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 12.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 98.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \left(\frac{\left|x\right|}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(\left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right), x\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + x \cdot \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right), x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot x\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot x\right)\right)\right), x\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \left(\frac{\frac{1}{x}}{x} \cdot x\right)\right)\right), x\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \frac{\frac{1}{x} \cdot x}{x}\right)\right), x\right) \]

      11. lft-mult-inverse N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right), x\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right) \]

   5. Simplified 98.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(x + \left|x\right|\right) + \frac{0.5}{x}\right)}, x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 97.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, x\right)\right), x\right) \]

   8. Simplified 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. Add Preprocessing

Alternative 17: 70.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -10.0)
   (copysign (log (- x)) x)
   (if (<= x 0.20000000298023224) (copysign x x) (copysign (log x) x))))

C

float code(float x) {
	float tmp;
	if (x <= -10.0f) {
		tmp = copysignf(logf(-x), x);
	} else if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-10.0))
		tmp = copysign(log(Float32(-x)), x);
	elseif (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-10.0))
		tmp = sign(x) * abs(log(-x));
	elseif (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 56.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f32 43.9%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{neg.f32}\left(x\right)\right), x\right) \]

   5. Simplified 43.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   if -10 < x < 0.200000003

   1. Initial program 21.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 94.6%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 94.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 12.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 98.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log x, x\right) \]

      4. log-lowering-log.f32 44.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   5. Simplified 44.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.20000000298023224) (copysign x x) (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.20000000298023224f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(0.20000000298023224))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(0.20000000298023224))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.200000003

   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

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

      2. fabs-lowering-fabs.f32 78.5%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

   5. Simplified 78.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Applied egg-rr 8.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 70.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 0.200000003 < x

   1. Initial program 61.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f32}\left(\log x, x\right) \]

      4. log-lowering-log.f32 44.2%

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(x\right), x\right) \]

   5. Simplified 44.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 53.6% 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 39.2%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
4. Step-by-step derivation

   1. accelerator-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), x\right) \]

   2. fabs-lowering-fabs.f32 70.3%

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), x\right) \]

5. Simplified 70.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

7. Applied egg-rr 7.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\frac{1 - x \cdot \left(x \cdot x\right)}{1 + \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + x \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x - x \cdot \left(x \cdot x\right)\right)}}}{1 - x \cdot x}\right)}, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{x}, x\right) \]
9. Step-by-step derivation

10. Simplified 56.4%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
11. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

float code(float x) {
	float t_0 = 1.0f / fabsf(x);
	return copysignf(log1pf((fabsf(x) + (fabsf(x) / (hypotf(1.0f, t_0) + t_0)))), x);
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) / abs(x))
	return copysign(log1p(Float32(abs(x) + Float32(abs(x) / Float32(hypot(Float32(1.0), t_0) + t_0)))), x)
end

Reproduce

herbie shell --seed 2024130 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (! :herbie-platform default (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))