Rust f32::asinh

Percentage Accurate: 37.4% → 76.7%

Time: 9.8s

Alternatives: 5

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Initial Program: 37.4% 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: 76.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.4%

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

      1. +-commutative 57.4%

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

      2. hypot-1-def 99.8%

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

      3. flip-+ 14.9%

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

      4. hypot-1-def 14.9%

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

      5. hypot-1-def 14.9%

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

      6. add-sqr-sqrt 14.9%

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

      7. +-commutative 14.9%

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

      8. hypot-1-def 14.9%

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

      9. +-commutative 14.9%

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

      10. div-sub 15.1%

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

   4. Applied egg-rr 17.4%

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

      1. div-sub 20.9%

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

      2. remove-double-neg 20.9%

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

      3. remove-double-neg 20.9%

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

      4. fma-undefine 20.8%

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

      5. unpow2 20.8%

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

      6. associate--r+ 54.6%

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

      7. +-inverses 99.8%

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

      8. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

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

   1. Initial program 34.0%

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

   3. Taylor expanded in x around 0 17.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(-0.041666666666666664 \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) + 0.001388888888888889 \cdot \left({x}^{2} \cdot \left(45 \cdot \frac{1}{1 + \left|x\right|} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}}\right)\right)\right)\right) + 0.5 \cdot \frac{1}{1 + \left|x\right|}\right)}, x\right) \]

   5. Simplified 67.4%

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

4. Final simplification 74.1%

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

Alternative 2: 69.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 38.8%

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

3. Taylor expanded in x around 0 30.8%

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

   1. log1p-define 69.7%

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

5. Simplified 69.7%

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

6. Final simplification 69.7%

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

Alternative 3: 75.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot \left(x \cdot -0.05 - 0.125\right) - 0.16666666666666666\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -5.0)
   (copysign (log (/ -0.5 x)) x)
   (copysign
    (*
     x
     (+
      1.0
      (* (pow x 2.0) (- (* x (- (* x -0.05) 0.125)) 0.16666666666666666))))
    x)))

C

float code(float x) {
	float tmp;
	if (x <= -5.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf((x * (1.0f + (powf(x, 2.0f) * ((x * ((x * -0.05f) - 0.125f)) - 0.16666666666666666f)))), 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);
	else
		tmp = copysign(Float32(x * Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * Float32(Float32(x * Float32(Float32(x * Float32(-0.05)) - Float32(0.125))) - Float32(0.16666666666666666))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-5.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	else
		tmp = sign(x) * abs((x * (single(1.0) + ((x ^ single(2.0)) * ((x * ((x * single(-0.05)) - single(0.125))) - single(0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5

   1. Initial program 54.2%

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

   3. Taylor expanded in x around -inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. distribute-rgt-neg-in 99.1%

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

      4. mul-1-neg 99.1%

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

      5. unsub-neg 99.1%

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

      6. sub-neg 99.1%

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

      7. rem-square-sqrt -0.0%

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

      8. fabs-sqr -0.0%

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

      9. rem-square-sqrt 12.5%

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

      10. associate-*r/ 12.5%

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

      11. metadata-eval 12.5%

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

      12. distribute-neg-frac 12.5%

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

      13. metadata-eval 12.5%

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

   5. Simplified 12.5%

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

   6. Taylor expanded in x around 0 97.6%

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

   if -5 < x

   1. Initial program 35.2%

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

      1. +-commutative 35.2%

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

      2. hypot-1-def 49.7%

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

      3. flip-+ 16.7%

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

      4. hypot-1-def 16.7%

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

      5. hypot-1-def 16.6%

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

      6. add-sqr-sqrt 16.6%

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

      7. +-commutative 16.6%

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

      8. hypot-1-def 16.6%

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

      9. +-commutative 16.6%

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

      10. div-sub 16.6%

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

   4. Applied egg-rr 16.4%

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

      1. div-sub 16.7%

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

      2. remove-double-neg 16.7%

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

      3. remove-double-neg 16.7%

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

      4. fma-undefine 16.7%

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

      5. unpow2 16.7%

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

      6. associate--r+ 16.8%

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

      7. +-inverses 16.8%

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

      8. metadata-eval 16.8%

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

   6. Simplified 16.8%

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

   7. Taylor expanded in x around 0 20.7%

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

   8. Taylor expanded in x around 0 66.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot \left(-0.05 \cdot x - 0.125\right) - 0.16666666666666666\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(x \cdot \left(x \cdot -0.05 - 0.125\right) - 0.16666666666666666\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 38.8%

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

3. Taylor expanded in x around 0 30.8%

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

   1. log1p-define 69.7%

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

   2. rem-square-sqrt 37.0%

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

   3. fabs-sqr 37.0%

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

   4. rem-square-sqrt 61.1%

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

5. Simplified 61.1%

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

6. Final simplification 61.1%

   \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right) \]
7. Add Preprocessing

Alternative 5: 55.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 38.8%

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

3. Taylor expanded in x around 0 20.2%

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

   1. +-commutative 20.2%

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

   2. fma-define 20.2%

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

   3. rem-square-sqrt 10.3%

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

   4. fabs-sqr 10.3%

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

   5. rem-square-sqrt 20.3%

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

5. Simplified 20.3%

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

6. Taylor expanded in x around 0 55.8%

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

   1. distribute-rgt-in 55.8%

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

   2. *-lft-identity 55.8%

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

   3. associate-*l* 55.8%

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

   4. unpow2 55.8%

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

   5. unpow3 55.8%

      \[\leadsto \mathsf{copysign}\left(x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}}, x\right) \]

8. Simplified 55.8%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x + -0.16666666666666666 \cdot {x}^{3}}, x\right) \]

9. Taylor expanded in x around 0 56.1%

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

10. Final simplification 56.1%

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

Developer target: 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 2024066 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))