Rust f32::asinh

Percentage Accurate: 38.1% → 99.1%

Time: 9.4s

Alternatives: 13

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 38.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ t_2 := \left|x\right| + \mathsf{hypot}\left(1, x\right)\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log t\_2, x\right)\\ \mathbf{elif}\;t\_0 \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-0.125}{t\_1} + \frac{-0.125}{{t\_1}^{2}}\right) + \frac{0.5}{t\_1}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{t\_2}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (+ (fabs x) 1.0))
        (t_2 (+ (fabs x) (hypot 1.0 x))))
   (if (<= t_0 -0.5)
     (copysign (log t_2) x)
     (if (<= t_0 0.0010000000474974513)
       (copysign
        (+
         (log1p (fabs x))
         (*
          x
          (*
           x
           (+
            (* (* x x) (+ (/ -0.125 t_1) (/ -0.125 (pow t_1 2.0))))
            (/ 0.5 t_1)))))
        x)
       (copysign (- (log (/ 1.0 t_2))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = fabsf(x) + 1.0f;
	float t_2 = fabsf(x) + hypotf(1.0f, x);
	float tmp;
	if (t_0 <= -0.5f) {
		tmp = copysignf(logf(t_2), x);
	} else if (t_0 <= 0.0010000000474974513f) {
		tmp = copysignf((log1pf(fabsf(x)) + (x * (x * (((x * x) * ((-0.125f / t_1) + (-0.125f / powf(t_1, 2.0f)))) + (0.5f / t_1))))), x);
	} else {
		tmp = copysignf(-logf((1.0f / t_2)), 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(abs(x) + Float32(1.0))
	t_2 = Float32(abs(x) + hypot(Float32(1.0), x))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.5))
		tmp = copysign(log(t_2), x);
	elseif (t_0 <= Float32(0.0010000000474974513))
		tmp = copysign(Float32(log1p(abs(x)) + Float32(x * Float32(x * Float32(Float32(Float32(x * x) * Float32(Float32(Float32(-0.125) / t_1) + Float32(Float32(-0.125) / (t_1 ^ Float32(2.0))))) + Float32(Float32(0.5) / t_1))))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / t_2))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 58.2%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 98.5%

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

   3. Simplified 98.5%

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

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.00100000005

   1. Initial program 17.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)}, x\right) \]
   4. Step-by-step derivation

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

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

      2. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), \left({x}^{2} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right|\right)\right), \left({x}^{2} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), \left({x}^{2} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), \left(x \cdot \left(x \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), \left(x \cdot \left(\left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right) \cdot x\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{+.f32}\left(\mathsf{log1p.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), \mathsf{*.f32}\left(x, \left(\left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right) \cdot x\right)\right)\right), x\right) \]

   5. Simplified 100.0%

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

   if 0.00100000005 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 52.2%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

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

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

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

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

   4. Applied egg-rr 98.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-0.125}{\left|x\right| + 1} + \frac{-0.125}{{\left(\left|x\right| + 1\right)}^{2}}\right) + \frac{0.5}{\left|x\right| + 1}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + \mathsf{hypot}\left(1, x\right)\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log t\_1, x\right)\\ \mathbf{elif}\;t\_0 \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{t\_1}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (+ (fabs x) (hypot 1.0 x))))
   (if (<= t_0 -0.5)
     (copysign (log t_1) x)
     (if (<= t_0 0.0010000000474974513)
       (copysign (log1p (+ (fabs x) (* (* x x) 0.5))) x)
       (copysign (- (log (/ 1.0 t_1))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = fabsf(x) + hypotf(1.0f, x);
	float tmp;
	if (t_0 <= -0.5f) {
		tmp = copysignf(logf(t_1), x);
	} else if (t_0 <= 0.0010000000474974513f) {
		tmp = copysignf(log1pf((fabsf(x) + ((x * x) * 0.5f))), x);
	} else {
		tmp = copysignf(-logf((1.0f / t_1)), 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(abs(x) + hypot(Float32(1.0), x))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.5))
		tmp = copysign(log(t_1), x);
	elseif (t_0 <= Float32(0.0010000000474974513))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(0.5)))), x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / t_1))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 58.2%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 98.5%

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

   3. Simplified 98.5%

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

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.00100000005

   1. Initial program 17.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 17.8%

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

   5. Simplified 17.8%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 0.00100000005 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 52.2%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

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

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

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

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

   4. Applied egg-rr 98.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))
   (if (<= t_0 -0.5)
     t_1
     (if (<= t_0 0.0010000000474974513)
       (copysign (log1p (+ (fabs x) (* (* x x) 0.5))) x)
       t_1))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	float tmp;
	if (t_0 <= -0.5f) {
		tmp = t_1;
	} else if (t_0 <= 0.0010000000474974513f) {
		tmp = copysignf(log1pf((fabsf(x) + ((x * x) * 0.5f))), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.5))
		tmp = t_1;
	elseif (t_0 <= Float32(0.0010000000474974513))
		tmp = copysign(log1p(Float32(abs(x) + Float32(Float32(x * x) * Float32(0.5)))), x);
	else
		tmp = t_1;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -0.5 or 0.00100000005 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 55.5%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 98.3%

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

   3. Simplified 98.3%

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

   if -0.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.00100000005

   1. Initial program 17.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 17.8%

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

   5. Simplified 17.8%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.6%

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

   3. Taylor expanded in x around -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. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) + x \cdot 1\right)\right)\right), x\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

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

      8. --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 56.9%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. associate--r- N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. unpow3 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. associate-/l* N/A

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

      13. div-sub N/A

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

   8. Simplified 97.8%

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

   if -2 < x < 1

   1. Initial program 19.4%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 18.6%

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

   5. Simplified 18.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 98.9%

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

   7. Applied egg-rr 98.9%

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

   if 1 < x

   1. Initial program 50.5%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

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

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

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

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f32 98.0%

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

   5. Simplified 98.0%

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

   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 98.0%

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

4. Final simplification 98.4%

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

Alternative 5: 98.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.6%

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

   3. Taylor expanded in x around -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. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x} + 1\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) + x \cdot 1\right)\right)\right), x\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(0 - \left(x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right) + x\right)\right)\right), x\right) \]

      6. associate--r+ N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\left(\left(0 - x \cdot \left(-1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right) - x\right)\right), x\right) \]

      7. neg-sub0 N/A

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

      8. --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 56.9%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. associate--r- N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. unpow3 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. associate-/l* N/A

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

      13. div-sub N/A

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

   8. Simplified 97.8%

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

   if -2 < x < 1

   1. Initial program 19.4%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 18.6%

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

   5. Simplified 18.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 98.9%

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

   7. Applied egg-rr 98.9%

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

   if 1 < x

   1. Initial program 50.5%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 97.7%

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

   5. Simplified 97.7%

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

4. Final simplification 98.4%

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

Alternative 6: 98.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 97.1%

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

   8. Simplified 97.1%

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

   if -2 < x < 1

   1. Initial program 19.4%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 18.6%

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

   5. Simplified 18.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 98.9%

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

   7. Applied egg-rr 98.9%

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

   if 1 < x

   1. Initial program 50.5%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 97.7%

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

   5. Simplified 97.7%

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

4. Final simplification 98.2%

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

Alternative 7: 97.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 97.1%

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

   8. Simplified 97.1%

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

   if -2 < x < 1

   1. Initial program 19.4%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 18.6%

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

   5. Simplified 18.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 98.9%

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

   7. Applied egg-rr 98.9%

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

   if 1 < x

   1. Initial program 50.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 96.5%

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

4. Final simplification 98.0%

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

Alternative 8: 95.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (log1p (fabs x)) x)
     (copysign (log (+ x (fabs x))) x))))

C

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 57.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 97.1%

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

   8. Simplified 97.1%

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

   if -2 < x < 1

   1. Initial program 19.4%

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

   3. Taylor expanded in x around 0

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

      1. log1p-define N/A

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

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

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

      3. fabs-lowering-fabs.f32 95.4%

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

   5. Simplified 95.4%

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

   if 1 < x

   1. Initial program 50.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 96.5%

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

4. Final simplification 96.1%

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

Alternative 9: 82.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < -2

   1. Initial program 57.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 97.1%

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

   8. Simplified 97.1%

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

   if -2 < x

   1. Initial program 27.7%

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

   3. Taylor expanded in x around 0

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

      1. log1p-define N/A

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

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

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

      3. fabs-lowering-fabs.f32 82.0%

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

   5. Simplified 82.0%

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

Alternative 10: 40.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999350456404 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -9.999999350456404e-39)
   (copysign (log (/ -0.5 x)) x)
   (copysign (log (/ 1.0 x)) x)))

C

float code(float x) {
	float tmp;
	if (x <= -9.999999350456404e-39f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf(logf((1.0f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999350456404e-39))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(log(Float32(Float32(1.0) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-9.999999350456404e-39))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	else
		tmp = sign(x) * abs(log((single(1.0) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999935e-39

   1. Initial program 39.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 53.2%

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

   5. Simplified 53.2%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 53.9%

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

   8. Simplified 53.9%

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

   if -9.99999935e-39 < x

   1. Initial program 30.6%

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

   3. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

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

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

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

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f32 39.0%

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

   5. Simplified 39.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

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

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

   7. Applied egg-rr 2.5%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f32 23.4%

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

   10. Simplified 23.4%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{x}\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 40.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999350456404 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -9.999999350456404e-39)
   (copysign (log (/ -0.5 x)) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -9.999999350456404e-39f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999350456404e-39))
		tmp = copysign(log(Float32(Float32(-0.5) / 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(-9.999999350456404e-39))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999935e-39

   1. Initial program 39.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 53.2%

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

   5. Simplified 53.2%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 53.9%

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

   8. Simplified 53.9%

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

   if -9.99999935e-39 < x

   1. Initial program 30.6%

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

   3. Taylor expanded in x around 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 23.4%

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

   5. Simplified 23.4%

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

Alternative 12: 27.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999350456404 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -9.999999350456404e-39)
   (copysign (log (- x)) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -9.999999350456404e-39f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999350456404e-39))
		tmp = copysign(log(Float32(-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(-9.999999350456404e-39))
		tmp = sign(x) * abs(log(-x));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999935e-39

   1. Initial program 39.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 27.1%

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

   5. Simplified 27.1%

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

   if -9.99999935e-39 < x

   1. Initial program 30.6%

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

   3. Taylor expanded in x around 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 23.4%

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

   5. Simplified 23.4%

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

Alternative 13: 13.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 35.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
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 11.8%

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

5. Simplified 11.8%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
6. 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 2024161 
(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))