?

Average Error: 20.5 → 0.2
Time: 10.4s
Precision: binary32
Cost: 9896

?

\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
↓
(FPCore (x)
 :precision binary32
 (if (<= x -0.5)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 0.10000000149011612)
     (copysign (log1p (+ x (/ (* x x) (+ 2.0 (* (* x x) 0.5))))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))
float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}
↓
float code(float x) {
	float tmp;
	if (x <= -0.5f) {
		tmp = copysignf(logf((1.0f / (hypotf(1.0f, x) - x))), x);
	} else if (x <= 0.10000000149011612f) {
		tmp = copysignf(log1pf((x + ((x * x) / (2.0f + ((x * x) * 0.5f))))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, x))), x);
	}
	return tmp;
}

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end
↓
function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.5))
		tmp = copysign(log(Float32(Float32(1.0) / Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.10000000149011612))
		tmp = copysign(log1p(Float32(x + Float32(Float32(x * x) / Float32(Float32(2.0) + Float32(Float32(x * x) * Float32(0.5)))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.5:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\

\mathbf{elif}\;x \leq 0.10000000149011612:\\
\;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\


\end{array}

Error?

Target

Original20.5
Target0.1
Herbie0.2
\[\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, \frac{1}{\left|x\right|}\right) + \frac{1}{\left|x\right|}}\right), x\right) \]

Derivation?

  1. Split input into 3 regimes

  2. if x < -0.5

    1. Initial program 15.5

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

    2. Simplified0.3

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

      [Start]15.5

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

      +-commutative [=>]15.5

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

      hypot-1-def [=>]0.3

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

    3. Applied egg-rr29.6

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

    4. Simplified0.3

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

      [Start]29.6

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

      div-sub [<=]28.9

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

      +-commutative [=>]28.9

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

      associate--r+ [=>]15.5

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

      +-inverses [=>]0.3

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

      metadata-eval [=>]0.3

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

      metadata-eval [<=]0.3

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

      associate-/r* [<=]0.3

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

      neg-mul-1 [<=]0.3

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

      neg-sub0 [=>]0.3

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

      associate--r- [=>]0.3

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

      neg-sub0 [<=]0.3

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

      mul-1-neg [<=]0.3

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

      +-commutative [<=]0.3

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

      mul-1-neg [=>]0.3

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

      sub-neg [<=]0.3

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

    if -0.5 < x < 0.100000001

    1. Initial program 25.2

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

    2. Simplified25.2

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

      [Start]25.2

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

      +-commutative [=>]25.2

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

      hypot-1-def [=>]25.2

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

    3. Applied egg-rr25.1

      \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} - 1}, x\right) \]

    4. Applied egg-rr0.9

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)}, x\right) \]

    5. Applied egg-rr0.0

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\left(x \cdot x + 0\right) \cdot \frac{1}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

    6. Simplified0.0

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

      [Start]0.0

      \[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x \cdot x + 0\right) \cdot \frac{1}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      associate-*r/ [=>]0.0

      \[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{\left(x \cdot x + 0\right) \cdot 1}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

      +-rgt-identity [=>]0.0

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

      *-rgt-identity [=>]0.0

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

    7. Taylor expanded in x around 0 0.1

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

    8. Simplified0.1

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{\color{blue}{2 + 0.5 \cdot \left(x \cdot x\right)}}\right), x\right) \]
      Proof

      [Start]0.1

      \[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + 0.5 \cdot {x}^{2}}\right), x\right) \]

      unpow2 [=>]0.1

      \[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}}\right), x\right) \]

    if 0.100000001 < x

    1. Initial program 16.2

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

    2. Simplified0.3

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

      [Start]16.2

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

      +-commutative [=>]16.2

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

      hypot-1-def [=>]0.3

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

    3. Applied egg-rr0.3

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

    4. Simplified0.3

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

      [Start]0.3

      \[ \mathsf{copysign}\left(0 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

      +-lft-identity [=>]0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.6
Cost23044
\[\begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x + \left(\mathsf{hypot}\left(1, x\right) - \left(x + x\right)\right)}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(-1 + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]
Alternative 2
Error0.3
Cost9896
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
Alternative 3
Error0.2
Cost9896
\[\begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
Alternative 4
Error0.6
Cost9892
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(-1 + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]
Alternative 5
Error0.5
Cost6984
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]
Alternative 6
Error0.6
Cost6792
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]
Alternative 7
Error0.5
Cost6792
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]
Alternative 8
Error0.7
Cost6760
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]
Alternative 9
Error5.2
Cost6664
\[\begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\frac{x \cdot \left(x + 2\right)}{x + 2}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]
Alternative 10
Error0.9
Cost6664
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]
Alternative 11
Error9.9
Cost6564
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
Alternative 12
Error12.0
Cost6532
\[\begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
Alternative 13
Error14.6
Cost3264
\[\mathsf{copysign}\left(x, x\right) \]

Error

Reproduce?

herbie shell --seed 2023083 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :herbie-target
  (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))