Rust f32::asinh

Percentage Accurate: 37.6% → 99.4%

Time: 8.9s

Alternatives: 18

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + \mathsf{hypot}\left(1, x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_2 := \left|x\right| + 1\\ t_3 := {t\_2}^{2}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{t\_0}\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \left(\frac{0.5}{t\_2} + x \cdot \left(x \cdot \left(\left(\frac{-0.125}{t\_2} + \frac{-0.125}{t\_3}\right) + 0.001388888888888889 \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{45}{t\_2} + \left(\frac{45}{t\_3} + \frac{30}{{t\_2}^{3}}\right)\right)\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log t\_0, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ (fabs x) (hypot 1.0 x)))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_2 (+ (fabs x) 1.0))
        (t_3 (pow t_2 2.0)))
   (if (<= t_1 -1.0)
     (copysign (- (log (/ 1.0 t_0))) x)
     (if (<= t_1 0.10000000149011612)
       (copysign
        (+
         (log1p (fabs x))
         (*
          (* x x)
          (+
           (/ 0.5 t_2)
           (*
            x
            (*
             x
             (+
              (+ (/ -0.125 t_2) (/ -0.125 t_3))
              (*
               0.001388888888888889
               (*
                (* x x)
                (+
                 (/ 45.0 t_2)
                 (+ (/ 45.0 t_3) (/ 30.0 (pow t_2 3.0))))))))))))
        x)
       (copysign (log t_0) x)))))

C

float code(float x) {
	float t_0 = fabsf(x) + hypotf(1.0f, x);
	float t_1 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_2 = fabsf(x) + 1.0f;
	float t_3 = powf(t_2, 2.0f);
	float tmp;
	if (t_1 <= -1.0f) {
		tmp = copysignf(-logf((1.0f / t_0)), x);
	} else if (t_1 <= 0.10000000149011612f) {
		tmp = copysignf((log1pf(fabsf(x)) + ((x * x) * ((0.5f / t_2) + (x * (x * (((-0.125f / t_2) + (-0.125f / t_3)) + (0.001388888888888889f * ((x * x) * ((45.0f / t_2) + ((45.0f / t_3) + (30.0f / powf(t_2, 3.0f)))))))))))), x);
	} else {
		tmp = copysignf(logf(t_0), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(abs(x) + hypot(Float32(1.0), x))
	t_1 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_2 = Float32(abs(x) + Float32(1.0))
	t_3 = t_2 ^ Float32(2.0)
	tmp = Float32(0.0)
	if (t_1 <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / t_0))), x);
	elseif (t_1 <= Float32(0.10000000149011612))
		tmp = copysign(Float32(log1p(abs(x)) + Float32(Float32(x * x) * Float32(Float32(Float32(0.5) / t_2) + Float32(x * Float32(x * Float32(Float32(Float32(Float32(-0.125) / t_2) + Float32(Float32(-0.125) / t_3)) + Float32(Float32(0.001388888888888889) * Float32(Float32(x * x) * Float32(Float32(Float32(45.0) / t_2) + Float32(Float32(Float32(45.0) / t_3) + Float32(Float32(30.0) / (t_2 ^ Float32(3.0))))))))))))), x);
	else
		tmp = copysign(log(t_0), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 44.1%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

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

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

   4. Applied egg-rr 97.0%

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

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

   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

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

   5. Simplified 100.0%

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

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

   1. Initial program 53.9%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 98.5%

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

   3. Simplified 98.5%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + \mathsf{hypot}\left(1, x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_2 := \left|x\right| + 1\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{t\_0}\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \left(\frac{0.5}{t\_2} + \left(x \cdot x\right) \cdot \left(\frac{-0.125}{t\_2} + \frac{-0.125}{{t\_2}^{2}}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log t\_0, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ (fabs x) (hypot 1.0 x)))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_2 (+ (fabs x) 1.0)))
   (if (<= t_1 -1.0)
     (copysign (- (log (/ 1.0 t_0))) x)
     (if (<= t_1 0.05000000074505806)
       (copysign
        (+
         (log1p (fabs x))
         (*
          (* x x)
          (+
           (/ 0.5 t_2)
           (* (* x x) (+ (/ -0.125 t_2) (/ -0.125 (pow t_2 2.0)))))))
        x)
       (copysign (log t_0) x)))))

C

float code(float x) {
	float t_0 = fabsf(x) + hypotf(1.0f, x);
	float t_1 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_2 = fabsf(x) + 1.0f;
	float tmp;
	if (t_1 <= -1.0f) {
		tmp = copysignf(-logf((1.0f / t_0)), x);
	} else if (t_1 <= 0.05000000074505806f) {
		tmp = copysignf((log1pf(fabsf(x)) + ((x * x) * ((0.5f / t_2) + ((x * x) * ((-0.125f / t_2) + (-0.125f / powf(t_2, 2.0f))))))), x);
	} else {
		tmp = copysignf(logf(t_0), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(abs(x) + hypot(Float32(1.0), x))
	t_1 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_2 = Float32(abs(x) + Float32(1.0))
	tmp = Float32(0.0)
	if (t_1 <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / t_0))), x);
	elseif (t_1 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(log1p(abs(x)) + Float32(Float32(x * x) * Float32(Float32(Float32(0.5) / t_2) + Float32(Float32(x * x) * Float32(Float32(Float32(-0.125) / t_2) + Float32(Float32(-0.125) / (t_2 ^ Float32(2.0)))))))), x);
	else
		tmp = copysign(log(t_0), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 44.1%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

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

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

   4. Applied egg-rr 97.0%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0500000007

   1. Initial program 23.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. log1p-define N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

   5. Simplified 99.9%

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

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 55.2%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \left(\frac{0.5}{\left|x\right| + 1} + \left(x \cdot x\right) \cdot \left(\frac{-0.125}{\left|x\right| + 1} + \frac{-0.125}{{\left(\left|x\right| + 1\right)}^{2}}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + \mathsf{hypot}\left(1, x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_1 \leq -0.02500000037252903:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{t\_0}\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.02500000037252903:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log t\_0, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ (fabs x) (hypot 1.0 x)))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_1 -0.02500000037252903)
     (copysign (- (log (/ 1.0 t_0))) x)
     (if (<= t_1 0.02500000037252903)
       (copysign (log1p (+ (fabs x) (* x (* x 0.5)))) x)
       (copysign (log t_0) x)))))

C

float code(float x) {
	float t_0 = fabsf(x) + hypotf(1.0f, x);
	float t_1 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_1 <= -0.02500000037252903f) {
		tmp = copysignf(-logf((1.0f / t_0)), x);
	} else if (t_1 <= 0.02500000037252903f) {
		tmp = copysignf(log1pf((fabsf(x) + (x * (x * 0.5f)))), x);
	} else {
		tmp = copysignf(logf(t_0), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float32(abs(x) + hypot(Float32(1.0), x))
	t_1 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_1 <= Float32(-0.02500000037252903))
		tmp = copysign(Float32(-log(Float32(Float32(1.0) / t_0))), x);
	elseif (t_1 <= Float32(0.02500000037252903))
		tmp = copysign(log1p(Float32(abs(x) + Float32(x * Float32(x * Float32(0.5))))), x);
	else
		tmp = copysign(log(t_0), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -0.0250000004

   1. Initial program 44.8%

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

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

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

   4. Applied egg-rr 96.8%

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

   if -0.0250000004 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0250000004

   1. Initial program 22.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f32 22.0%

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

   5. Simplified 22.0%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 0.0250000004 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 56.2%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 98.2%

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

   3. Simplified 98.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.02500000037252903:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.02500000037252903:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.02500000037252903:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -1.0)
     t_1
     (if (<= t_0 0.02500000037252903)
       (copysign (log1p (+ (fabs x) (* x (* x 0.5)))) x)
       t_1))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float t_1 = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = t_1;
	} else if (t_0 <= 0.02500000037252903f) {
		tmp = copysignf(log1pf((fabsf(x) + (x * (x * 0.5f)))), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	t_1 = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = t_1;
	elseif (t_0 <= Float32(0.02500000037252903))
		tmp = copysign(log1p(Float32(abs(x) + Float32(x * Float32(x * Float32(0.5))))), x);
	else
		tmp = t_1;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < -1 or 0.0250000004 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 50.5%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f32 97.6%

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

   3. Simplified 97.6%

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

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 0.0250000004

   1. Initial program 22.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f32 22.5%

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

   5. Simplified 22.5%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 99.6%

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.02500000037252903:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.5)
     (copysign (log1p (+ (fabs x) (* x (* x 0.5)))) x)
     (copysign
      (log
       (*
        x
        (+
         (+ (/ 0.5 (* x x)) (+ (/ (fabs x) x) (/ -0.125 (* x (* x (* x x))))))
         1.0)))
      x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf(log1pf((fabsf(x) + (x * (x * 0.5f)))), x);
	} else {
		tmp = copysignf(logf((x * (((0.5f / (x * x)) + ((fabsf(x) / x) + (-0.125f / (x * (x * (x * x)))))) + 1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / 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 * Float32(Float32(Float32(Float32(0.5) / Float32(x * x)) + Float32(Float32(abs(x) / x) + Float32(Float32(-0.125) / Float32(x * Float32(x * Float32(x * x)))))) + Float32(1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 43.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 96.3%

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

   5. Simplified 96.3%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 97.4%

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

   8. Simplified 97.4%

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

   if -5 < x < 0.5

   1. Initial program 26.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f32 24.7%

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

   5. Simplified 24.7%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 97.7%

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

   7. Applied egg-rr 97.7%

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

   if 0.5 < x

   1. Initial program 52.5%

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

   3. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate--l+ N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. sub-neg N/A

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

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

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

   5. Simplified 98.4%

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

4. Final simplification 97.8%

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

Alternative 6: 98.3% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.5)
     (copysign (log1p (+ (fabs x) (* x (* x 0.5)))) x)
     (copysign
      (log (* x (+ (/ (+ (fabs x) (/ (+ 0.5 (/ -0.125 (* x x))) x)) x) 1.0)))
      x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf(log1pf((fabsf(x) + (x * (x * 0.5f)))), x);
	} else {
		tmp = copysignf(logf((x * (((fabsf(x) + ((0.5f + (-0.125f / (x * x))) / x)) / x) + 1.0f))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / 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 * Float32(Float32(Float32(abs(x) + Float32(Float32(Float32(0.5) + Float32(Float32(-0.125) / Float32(x * x))) / x)) / x) + Float32(1.0)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 43.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 96.3%

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

   5. Simplified 96.3%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 97.4%

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

   8. Simplified 97.4%

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

   if -5 < x < 0.5

   1. Initial program 26.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f32 24.7%

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

   5. Simplified 24.7%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 97.7%

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

   7. Applied egg-rr 97.7%

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

   if 0.5 < x

   1. Initial program 52.5%

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

   3. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate--l+ N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. sub-neg N/A

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

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

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

   5. Simplified 98.4%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   8. Simplified 98.4%

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

4. Final simplification 97.8%

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

Alternative 7: 98.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.5)
     (copysign (log1p (+ (fabs x) (* x (* x 0.5)))) x)
     (copysign (log (+ (fabs x) (* x (+ (/ 0.5 (* x x)) 1.0)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 0.5f) {
		tmp = copysignf(log1pf((fabsf(x) + (x * (x * 0.5f)))), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + (x * ((0.5f / (x * x)) + 1.0f)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / 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(Float32(0.5) / Float32(x * x)) + Float32(1.0))))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 43.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 96.3%

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

   5. Simplified 96.3%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 97.4%

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

   8. Simplified 97.4%

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

   if -5 < x < 0.5

   1. Initial program 26.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f32 24.7%

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

   5. Simplified 24.7%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 97.7%

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

   7. Applied egg-rr 97.7%

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

   if 0.5 < x

   1. Initial program 52.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right), x\right) \]
   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 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 97.7%

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

Alternative 8: 97.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 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 <= -5.0f) {
		tmp = copysignf(logf((-0.5f / 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(-5.0))
		tmp = copysign(log(Float32(Float32(-0.5) / 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 < -5

   1. Initial program 43.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 96.3%

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

   5. Simplified 96.3%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 97.4%

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

   8. Simplified 97.4%

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

   if -5 < x < 0.5

   1. Initial program 26.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f32 24.7%

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

   5. Simplified 24.7%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f32 97.7%

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

   7. Applied egg-rr 97.7%

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

   if 0.5 < x

   1. Initial program 52.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \color{blue}{x}\right)\right), x\right) \]
   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 97.4%

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

Alternative 9: 95.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -1.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.5)
     (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((-0.5f / x)), x);
	} else if (x <= 0.5f) {
		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(Float32(-0.5) / x)), x);
	elseif (x <= Float32(0.5))
		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 44.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 95.5%

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

   5. Simplified 95.5%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 96.5%

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

   8. Simplified 96.5%

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

   if -1 < x < 0.5

   1. Initial program 25.8%

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

   3. Taylor expanded in x around 0

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

      1. log1p-define N/A

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

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

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

      3. fabs-lowering-fabs.f32 92.0%

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

   5. Simplified 92.0%

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

   if 0.5 < x

   1. Initial program 52.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{fabs.f32}\left(x\right), \color{blue}{x}\right)\right), x\right) \]
   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 94.2%

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

Alternative 10: 82.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 44.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 95.5%

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

   5. Simplified 95.5%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 96.5%

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

   8. Simplified 96.5%

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

   if -1 < x

   1. Initial program 34.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 76.5%

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

   5. Simplified 76.5%

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

Alternative 11: 42.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000004306106665 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-0.5 + \left(\log x + 1\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0000004306106665e-39)
   (copysign (log (/ -0.5 x)) x)
   (copysign (+ -0.5 (+ (log x) 1.0)) x)))

C

float code(float x) {
	float tmp;
	if (x <= -2.0000004306106665e-39f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf((-0.5f + (logf(x) + 1.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000004306106665e-39))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(Float32(Float32(-0.5) + Float32(log(x) + Float32(1.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0000004306106665e-39))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	else
		tmp = sign(x) * abs((single(-0.5) + (log(x) + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.0000004e-39

   1. Initial program 34.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 53.4%

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

   5. Simplified 53.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 54.8%

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

   8. Simplified 54.8%

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

   if -2.0000004e-39 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{1 + -1 \cdot {\left(\left|x\right|\right)}^{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 27.0%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

      4. clear-num N/A

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

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

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

      6. div-inv N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. sqr-abs N/A

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

      13. pow2 N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. inv-pow N/A

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

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

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

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

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

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

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

   7. Applied egg-rr 27.0%

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

      1. rgt-mult-inverse N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

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

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

      6. div-inv N/A

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

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

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

      8. associate-/r* N/A

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

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

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

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

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

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

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

      12. log-lowering-log.f32 27.0%

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

   9. Applied egg-rr 27.0%

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

   10. Taylor expanded in x around inf

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

   12. Simplified 29.1%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.5} + \left(\log x + 1\right), x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 42.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000004306106665 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(0.5 + \log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0000004306106665e-39)
   (copysign (log (/ -0.5 x)) x)
   (copysign (+ 0.5 (log x)) x)))

C

float code(float x) {
	float tmp;
	if (x <= -2.0000004306106665e-39f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf((0.5f + logf(x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000004306106665e-39))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(Float32(Float32(0.5) + log(x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0000004306106665e-39))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	else
		tmp = sign(x) * abs((single(0.5) + log(x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.0000004e-39

   1. Initial program 34.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 53.4%

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

   5. Simplified 53.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 54.8%

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

   8. Simplified 54.8%

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

   if -2.0000004e-39 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{1 + -1 \cdot {\left(\left|x\right|\right)}^{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 27.0%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

      4. clear-num N/A

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

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

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

      6. div-inv N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. sqr-abs N/A

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

      13. pow2 N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. inv-pow N/A

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

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

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

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

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

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

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

   7. Applied egg-rr 27.0%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-lowering-log.f32 29.1%

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

   10. Simplified 29.1%

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

Alternative 13: 40.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000004306106665 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0000004306106665e-39)
   (copysign (log (/ -0.5 x)) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -2.0000004306106665e-39f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000004306106665e-39))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0000004306106665e-39))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.0000004e-39

   1. Initial program 34.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. unpow2 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 53.4%

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

   5. Simplified 53.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f32 54.8%

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

   8. Simplified 54.8%

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

   if -2.0000004e-39 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. log-lowering-log.f32 26.3%

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

   5. Simplified 26.3%

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

Alternative 14: 27.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000004306106665 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0000004306106665e-39)
   (copysign (log (- x)) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -2.0000004306106665e-39f) {
		tmp = copysignf(logf(-x), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000004306106665e-39))
		tmp = copysign(log(Float32(-x)), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0000004306106665e-39))
		tmp = sign(x) * abs(log(-x));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.0000004e-39

   1. Initial program 34.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 27.8%

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

   5. Simplified 27.8%

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

   if -2.0000004e-39 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. log-lowering-log.f32 26.3%

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

   5. Simplified 26.3%

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

Alternative 15: 17.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000004306106665 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\frac{0.5}{x}}{x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0000004306106665e-39)
   (copysign (/ (/ 0.5 x) x) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (x <= -2.0000004306106665e-39f) {
		tmp = copysignf(((0.5f / x) / x), x);
	} else {
		tmp = copysignf(logf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000004306106665e-39))
		tmp = copysign(Float32(Float32(Float32(0.5) / x) / x), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0000004306106665e-39))
		tmp = sign(x) * abs(((single(0.5) / x) / x));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.0000004e-39

   1. Initial program 34.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{1 + -1 \cdot {\left(\left|x\right|\right)}^{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 7.2%

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

   8. Simplified 7.2%

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

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

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f32 7.2%

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

   10. Applied egg-rr 7.2%

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

   if -2.0000004e-39 < x

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. log-lowering-log.f32 26.3%

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

   5. Simplified 26.3%

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

Alternative 16: 7.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{1 + -1 \cdot {\left(\left|x\right|\right)}^{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
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 14.3%

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

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 7.2%

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

8. Simplified 7.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{0.5}{x \cdot x}}, x\right) \]
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. div-inv N/A

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

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

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

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

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

   5. /-lowering-/.f32 7.2%

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

10. Applied egg-rr 7.2%

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

Alternative 17: 7.2% 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 36.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{1 + -1 \cdot {\left(\left|x\right|\right)}^{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
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 14.3%

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

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 7.2%

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

8. Simplified 7.2%

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

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

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

   2. associate-/r* N/A

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

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

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

   4. /-lowering-/.f32 7.2%

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

10. Applied egg-rr 7.2%

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

Alternative 18: 7.2% 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 36.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign.f32}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{1 + -1 \cdot {\left(\left|x\right|\right)}^{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
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 14.3%

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

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 7.2%

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

8. Simplified 7.2%

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

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Reproduce

herbie shell --seed 2024138 
(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))