Rust f32::asinh

Percentage Accurate: 37.6% → 99.5%
Time: 3.2s
Alternatives: 2
Speedup: 2.2×

Specification

?
\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]
(FPCore (x) :precision binary32 (asinh x))
float code(float x) {
	return asinhf(x);
}

function code(x)
	return asinh(x)
end

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

\begin{array}{l}

\\
\sinh^{-1} x
\end{array}

Sampling outcomes in binary32 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 37.6% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \end{array} \]
(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

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

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

\begin{array}{l}

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

Alternative 1: 99.5% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \mathsf{copysign}\left(\sinh^{-1} x, x\right) \end{array} \]
(FPCore (x) :precision binary32 (copysign (asinh x) x))
float code(float x) {
	return copysignf(asinhf(x), x);
}

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

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

\begin{array}{l}

\\
\mathsf{copysign}\left(\sinh^{-1} x, x\right)
\end{array}
Derivation

  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. Step-by-step derivation

    1. Applied rewrites99.5%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]
    2. Add Preprocessing

    Alternative 2: 54.4% accurate, 2.2× speedup?

    \[\begin{array}{l} \\ \mathsf{copysign}\left(x, x\right) \end{array} \]
    (FPCore (x) :precision binary32 (copysign x x))
    float code(float x) {
	return copysignf(x, x);
}

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

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

    \begin{array}{l}

\\
\mathsf{copysign}\left(x, x\right)
\end{array}
    Derivation

    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 0

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

    5. Applied rewrites49.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), \mathsf{fma}\left(45, {\left(1 + x\right)}^{-2}, \frac{30}{{\left(1 + x\right)}^{3}}\right) + \frac{45}{1 + x}, \mathsf{fma}\left({\left(1 + x\right)}^{-2}, 3, \frac{3}{1 + x}\right) \cdot -0.041666666666666664\right), x \cdot x, \frac{0.5}{1 + x}\right), x \cdot x, \mathsf{log1p}\left(x\right)\right)}, x\right) \]

    6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(x, x\right) \]
    7. Step-by-step derivation

      1. Applied rewrites52.4%

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

      Developer Target 1: 99.7% accurate, 0.6× speedup?

      \[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]
      (FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))
      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);
}

      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

      \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}

      Reproduce

      ?
      herbie shell --seed 2025054 
(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))