Rust f32::asinh

Percentage Accurate: 38.6% → 99.7%

Time: 8.7s

Alternatives: 15

Speedup: 1.4×

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: 38.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

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

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 (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))
   (if (<= t_0 -0.20000000298023224)
     t_2
     (if (<= t_0 0.20000000298023224)
       (copysign
        (+
         (log1p (fabs x))
         (*
          (* x x)
          (+
           (/ 0.5 t_1)
           (+
            (* (/ -0.125 t_1) (+ (* x x) (/ (* x x) t_1)))
            (*
             (+ (/ 45.0 (pow t_1 2.0)) (+ (/ 45.0 t_1) (/ 30.0 (pow t_1 3.0))))
             (* x (* x (* (* x x) 0.001388888888888889))))))))
        x)
       t_2))))

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 = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	float tmp;
	if (t_0 <= -0.20000000298023224f) {
		tmp = t_2;
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf((log1pf(fabsf(x)) + ((x * x) * ((0.5f / t_1) + (((-0.125f / t_1) * ((x * x) + ((x * x) / t_1))) + (((45.0f / powf(t_1, 2.0f)) + ((45.0f / t_1) + (30.0f / powf(t_1, 3.0f)))) * (x * (x * ((x * x) * 0.001388888888888889f)))))))), x);
	} else {
		tmp = t_2;
	}
	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 = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.20000000298023224))
		tmp = t_2;
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(log1p(abs(x)) + Float32(Float32(x * x) * Float32(Float32(Float32(0.5) / t_1) + Float32(Float32(Float32(Float32(-0.125) / t_1) * Float32(Float32(x * x) + Float32(Float32(x * x) / t_1))) + Float32(Float32(Float32(Float32(45.0) / (t_1 ^ Float32(2.0))) + Float32(Float32(Float32(45.0) / t_1) + Float32(Float32(30.0) / (t_1 ^ Float32(3.0))))) * Float32(x * Float32(x * Float32(Float32(x * x) * Float32(0.001388888888888889))))))))), x);
	else
		tmp = t_2;
	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) < -0.200000003 or 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 50.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 99.1%

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

   3. Simplified 99.1%

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

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

   1. Initial program 22.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right) + \left|x\right|\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

   5. Simplified 22.7%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(\frac{-0.125}{\left|x\right| + 1} \cdot \left(x \cdot x + \frac{x \cdot x}{\left|x\right| + 1}\right) + \left(\frac{45}{{\left(\left|x\right| + 1\right)}^{2}} + \left(\frac{45}{\left|x\right| + 1} + \frac{30}{{\left(\left|x\right| + 1\right)}^{3}}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.20000000298023224:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))
   (if (<= t_0 -0.20000000298023224)
     t_1
     (if (<= t_0 0.20000000298023224)
       (copysign
        (log1p
         (+
          (fabs x)
          (* x (* x (+ 0.5 (* x (* x (+ -0.125 (* (* x x) 0.0625)))))))))
        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 <= -0.20000000298023224f) {
		tmp = t_1;
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(log1pf((fabsf(x) + (x * (x * (0.5f + (x * (x * (-0.125f + ((x * x) * 0.0625f))))))))), 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(-0.20000000298023224))
		tmp = t_1;
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(log1p(Float32(abs(x) + Float32(x * Float32(x * Float32(Float32(0.5) + Float32(x * Float32(x * Float32(Float32(-0.125) + Float32(Float32(x * x) * Float32(0.0625)))))))))), 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) < -0.200000003 or 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 50.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 99.1%

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

   3. Simplified 99.1%

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

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

   1. Initial program 22.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right) + \left|x\right|\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

   5. Simplified 22.7%

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

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right) + \left|x\right|\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right) + \left|x\right|\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \left(\left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      14. *-lowering-*.f32 99.9%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(-1 + \frac{\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x \cdot \left(1 + \frac{0.5}{x \cdot x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign
    (log (* x (+ -1.0 (/ (+ (fabs x) (/ (+ -0.5 (/ 0.125 (* x x))) x)) x))))
    x)
   (if (<= x 0.5)
     (copysign
      (log1p
       (+
        (fabs x)
        (* x (* x (+ 0.5 (* x (* x (+ -0.125 (* (* x x) 0.0625)))))))))
      x)
     (copysign (log (+ (fabs x) (* x (+ 1.0 (/ 0.5 (* x x)))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((x * (-1.0f + ((fabsf(x) + ((-0.5f + (0.125f / (x * x))) / x)) / x)))), x);
	} else if (x <= 0.5f) {
		tmp = copysignf(log1pf((fabsf(x) + (x * (x * (0.5f + (x * (x * (-0.125f + ((x * x) * 0.0625f))))))))), 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(-1.0))
		tmp = copysign(log(Float32(x * Float32(Float32(-1.0) + Float32(Float32(abs(x) + Float32(Float32(Float32(-0.5) + Float32(Float32(0.125) / Float32(x * x))) / x)) / x)))), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(log1p(Float32(abs(x) + Float32(x * Float32(x * Float32(Float32(0.5) + Float32(x * Float32(x * Float32(Float32(-0.125) + Float32(Float32(x * x) * Float32(0.0625)))))))))), 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 < -1

   1. Initial program 44.7%

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

   3. Taylor expanded in x around -inf

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

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

   5. Simplified 98.8%

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

   if -1 < x < 0.5

   1. Initial program 24.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right) + \left|x\right|\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

   5. Simplified 24.0%

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

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right) + \left|x\right|\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right) + \left|x\right|\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \left(\left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      14. *-lowering-*.f32 99.0%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)\right)}, x\right) \]

   if 0.5 < x

   1. Initial program 51.3%

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

   3. Taylor expanded in x around 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 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(-1 + \frac{\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x \cdot \left(1 + \frac{0.5}{x \cdot x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((x * (-1.0f + ((fabsf(x) + ((-0.5f + (0.125f / (x * x))) / x)) / x)))), x);
	} else if (x <= 0.5f) {
		tmp = copysignf(log1pf((fabsf(x) + (x * (x * (0.5f + ((x * x) * -0.125f)))))), 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(-1.0))
		tmp = copysign(log(Float32(x * Float32(Float32(-1.0) + Float32(Float32(abs(x) + Float32(Float32(Float32(-0.5) + Float32(Float32(0.125) / Float32(x * x))) / x)) / x)))), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(log1p(Float32(abs(x) + Float32(x * Float32(x * Float32(Float32(0.5) + Float32(Float32(x * x) * Float32(-0.125))))))), 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 < -1

   1. Initial program 44.7%

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

   3. Taylor expanded in x around -inf

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

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

   5. Simplified 98.8%

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

   if -1 < x < 0.5

   1. Initial program 24.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right) + \left|x\right|\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

   5. Simplified 24.0%

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

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right) + \left|x\right|\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right) + \left|x\right|\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \left(\left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      14. *-lowering-*.f32 99.0%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\color{blue}{\left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   9. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 98.5%

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

   10. Simplified 98.5%

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

   if 0.5 < x

   1. Initial program 51.3%

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

   3. Taylor expanded in x around 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 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 98.5%

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

Alternative 5: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x \cdot \left(1 + \frac{0.5}{x \cdot x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 0.5)
     (copysign (log1p (+ (fabs x) (* x (* x (+ 0.5 (* (* x x) -0.125)))))) x)
     (copysign (log (+ (fabs x) (* x (+ 1.0 (/ 0.5 (* x x)))))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf(log1pf((fabsf(x) + (x * (x * (0.5f + ((x * x) * -0.125f)))))), 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(-50.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(log1p(Float32(abs(x) + Float32(x * Float32(x * Float32(Float32(0.5) + Float32(Float32(x * x) * Float32(-0.125))))))), 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 < -50

   1. Initial program 43.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 99.5%

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

   5. Simplified 99.5%

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

   if -50 < x < 0.5

   1. Initial program 25.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right) + \left|x\right|\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \mathsf{+.f32}\left(\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right), \left(\left|x\right|\right)\right)\right)\right), x\right) \]

   5. Simplified 24.2%

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

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f32}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right) + \left|x\right|\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right) + \left|x\right|\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\left(\left|x\right| + x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\left|x\right|\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{-1}{8} + \left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \left(\left(x \cdot x\right) \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      14. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 98.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log1p.f32}\left(\color{blue}{\left(\left|x\right| + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   9. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 98.1%

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

   10. Simplified 98.1%

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

   if 0.5 < x

   1. Initial program 51.3%

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

   3. Taylor expanded in x around 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 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x \cdot \left(1 + \frac{0.5}{x \cdot x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x \cdot \left(1 + \frac{0.5}{x \cdot x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 0.5)
     (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 <= -50.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 0.5f) {
		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(-50.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(log1p(Float32(abs(x) + Float32(x * Float32(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 < -50

   1. Initial program 43.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 99.5%

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

   5. Simplified 99.5%

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

   if -50 < x < 0.5

   1. Initial program 25.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right), x\right) \]
   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 22.4%

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

   5. Simplified 22.4%

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

   6. Taylor expanded in x around 0

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

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

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

      2. log1p-define N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f32 96.5%

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

   8. Simplified 96.5%

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

   if 0.5 < x

   1. Initial program 51.3%

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

   3. Taylor expanded in x around 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 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x \cdot \left(1 + \frac{0.5}{x \cdot x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -50.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 0.5)
     (copysign (log1p (+ (fabs x) (* x (* x 0.5)))) x)
     (copysign (log (+ x (fabs x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -50.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 0.5f) {
		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(-50.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (x <= Float32(0.5))
		tmp = copysign(log1p(Float32(abs(x) + Float32(x * Float32(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 < -50

   1. Initial program 43.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 99.5%

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

   5. Simplified 99.5%

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

   if -50 < x < 0.5

   1. Initial program 25.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right)\right), x\right) \]
   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 22.4%

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

   5. Simplified 22.4%

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

   6. Taylor expanded in x around 0

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

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

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

      2. log1p-define N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f32 96.5%

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

   8. Simplified 96.5%

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

   if 0.5 < x

   1. Initial program 51.3%

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

   3. Taylor expanded in x around 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 96.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 0.20000000298023224)
     (copysign (log1p (fabs x)) x)
     (copysign (log (+ x (fabs x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 0.20000000298023224f) {
		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(-1.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (x <= Float32(0.20000000298023224))
		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 < -1

   1. Initial program 44.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 98.0%

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

   5. Simplified 98.0%

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

   if -1 < x < 0.200000003

   1. Initial program 24.2%

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

   3. Taylor expanded in x around 0

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

      1. log1p-define N/A

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

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

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

      3. fabs-lowering-fabs.f32 91.9%

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

   5. Simplified 91.9%

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

   if 0.200000003 < x

   1. Initial program 51.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 96.0%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 0.20000000298023224)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ x (fabs x))) x)))

C

float code(float x) {
	float tmp;
	if (x <= 0.20000000298023224f) {
		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(0.20000000298023224))
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float32(x + abs(x))), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.200000003

   1. Initial program 29.1%

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

   3. Taylor expanded in x around 0

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

      1. log1p-define N/A

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

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

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

      3. fabs-lowering-fabs.f32 80.7%

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

   5. Simplified 80.7%

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

   if 0.200000003 < x

   1. Initial program 51.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 96.0%

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

4. Final simplification 85.0%

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

Alternative 10: 68.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (log1p (fabs x)) x))

C

float code(float x) {
	return copysignf(log1pf(fabsf(x)), x);
}

Julia

function code(x)
	return copysign(log1p(abs(x)), x)
end

Derivation

1. Initial program 35.5%

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

3. Taylor expanded in x around 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 70.4%

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

5. Simplified 70.4%

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

Alternative 11: 27.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -4.999999955487895e-38)
   (copysign (log (- x)) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -4.999999955487895e-38f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999955487895e-38))
		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(-4.999999955487895e-38))
		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 < -4.99999996e-38

   1. Initial program 31.0%

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

   3. Taylor expanded in x around -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 23.6%

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

   5. Simplified 23.6%

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

   if -4.99999996e-38 < x

   1. Initial program 39.3%

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

   3. Taylor expanded in x around 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 27.9%

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

   5. Simplified 27.9%

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

Alternative 12: 17.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{copysign}\left(\frac{1}{x} \cdot \frac{0.5}{x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -4.999999955487895e-38)
   (copysign (* (/ 1.0 x) (/ 0.5 x)) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -4.999999955487895e-38f) {
		tmp = copysignf(((1.0f / x) * (0.5f / x)), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999955487895e-38))
		tmp = copysign(Float32(Float32(Float32(1.0) / x) * 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(-4.999999955487895e-38))
		tmp = sign(x) * abs(((single(1.0) / x) * (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 < -4.99999996e-38

   1. Initial program 31.0%

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

   3. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. unpow2 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. cancel-sign-sub-inv N/A

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

      15. sqr-abs N/A

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

      16. unpow2 N/A

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

      17. div-sub N/A

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

   5. Simplified -0.0%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 7.5%

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

   8. Simplified 7.5%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

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

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

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

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

      5. /-lowering-/.f32 7.5%

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

   10. Applied egg-rr 7.5%

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

   if -4.99999996e-38 < x

   1. Initial program 39.3%

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

   3. Taylor expanded in x around 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 27.9%

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

   5. Simplified 27.9%

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

Alternative 13: 7.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\frac{1}{x} \cdot \frac{0.5}{x}, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (* (/ 1.0 x) (/ 0.5 x)) x))

C

float code(float x) {
	return copysignf(((1.0f / x) * (0.5f / x)), x);
}

Julia

function code(x)
	return copysign(Float32(Float32(Float32(1.0) / x) * Float32(Float32(0.5) / x)), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(((single(1.0) / x) * (single(0.5) / x)));
end

Derivation

1. Initial program 35.5%

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

3. Taylor expanded in x around inf

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

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

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. remove-double-neg N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

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

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

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

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

   11. mul-1-neg N/A

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

   12. unpow2 N/A

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

   13. distribute-lft-neg-in N/A

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

   14. cancel-sign-sub-inv N/A

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

   15. sqr-abs N/A

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

   16. unpow2 N/A

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

   17. div-sub N/A

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

5. Simplified 15.7%

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

6. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 7.4%

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

8. Simplified 7.4%

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

   1. metadata-eval N/A

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

   2. frac-times N/A

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

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

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

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

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

   5. /-lowering-/.f32 7.4%

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

10. Applied egg-rr 7.4%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{1}{x} \cdot \frac{0.5}{x}}, x\right) \]
11. Add Preprocessing

Alternative 14: 7.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\frac{\frac{0.5}{x}}{x}, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (/ (/ 0.5 x) x) x))

C

float code(float x) {
	return copysignf(((0.5f / x) / x), x);
}

Julia

function code(x)
	return copysign(Float32(Float32(Float32(0.5) / x) / x), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(((single(0.5) / x) / x));
end

Derivation

1. Initial program 35.5%

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

3. Taylor expanded in x around inf

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

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

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. remove-double-neg N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

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

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

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

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

   11. mul-1-neg N/A

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

   12. unpow2 N/A

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

   13. distribute-lft-neg-in N/A

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

   14. cancel-sign-sub-inv N/A

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

   15. sqr-abs N/A

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

   16. unpow2 N/A

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

   17. div-sub N/A

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

5. Simplified 15.7%

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

6. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 7.4%

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

8. Simplified 7.4%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f32 7.4%

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

10. Applied egg-rr 7.4%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{\frac{0.5}{x}}{x}}, x\right) \]
11. Add Preprocessing

Alternative 15: 7.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\frac{0.5}{x \cdot x}, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (/ 0.5 (* x x)) x))

C

float code(float x) {
	return copysignf((0.5f / (x * x)), x);
}

Julia

function code(x)
	return copysign(Float32(Float32(0.5) / Float32(x * x)), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((single(0.5) / (x * x)));
end

Derivation

1. Initial program 35.5%

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

3. Taylor expanded in x around inf

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

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

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. remove-double-neg N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

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

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

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

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

   11. mul-1-neg N/A

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

   12. unpow2 N/A

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

   13. distribute-lft-neg-in N/A

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

   14. cancel-sign-sub-inv N/A

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

   15. sqr-abs N/A

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

   16. unpow2 N/A

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

   17. div-sub N/A

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

5. Simplified 15.7%

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

6. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 7.4%

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

8. Simplified 7.4%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{0.5}{x \cdot x}}, x\right) \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024164 
(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))