Rust f64::asinh

Percentage Accurate: 29.5% → 99.9%
Time: 1.4s
Alternatives: 2
Speedup: 1.5×

Specification

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\[\sinh^{-1} x \]
(FPCore (x)
  :precision binary64
  (asinh x))
double code(double x) {
	return asinh(x);
}

def code(x):
	return math.asinh(x)

function code(x)
	return asinh(x)
end

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

code[x_] := N[ArcSinh[x], $MachinePrecision]

\sinh^{-1} x

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: 29.5% accurate, 1.0× speedup?

\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
(FPCore (x)
  :precision binary64
  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
double code(double x) {
	return copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
}

public static double code(double x) {
	return Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
}

def code(x):
	return math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)

function code(x)
	return copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
end

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

code[x_] := N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

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

Alternative 1: 99.9% accurate, 1.5× speedup?

\[\mathsf{copysign}\left(\sinh^{-1} \left(\left|x\right|\right), x\right) \]
(FPCore (x)
  :precision binary64
  (copysign (asinh (fabs x)) x))
double code(double x) {
	return copysign(asinh(fabs(x)), x);
}

def code(x):
	return math.copysign(math.asinh(math.fabs(x)), x)

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

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

code[x_] := N[With[{TMP1 = Abs[N[ArcSinh[N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

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

  1. Initial program 29.5%

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

    1. lift-log.f64N/A

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

    2. lift-+.f64N/A

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

    3. lift-sqrt.f64N/A

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

    4. lift-+.f64N/A

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

    5. lift-*.f64N/A

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

    6. sqr-abs-revN/A

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

    7. lift-fabs.f64N/A

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

    8. lift-fabs.f64N/A

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

    9. asinh-def-revN/A

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

    10. lower-asinh.f6499.9%

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

  3. Applied rewrites99.9%

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

Alternative 2: 19.3% accurate, 1.6× speedup?

\[\mathsf{copysign}\left(\log \left(1 + \left|x\right|\right), x\right) \]
(FPCore (x)
  :precision binary64
  (copysign (log (+ 1.0 (fabs x))) x))
double code(double x) {
	return copysign(log((1.0 + fabs(x))), x);
}

public static double code(double x) {
	return Math.copySign(Math.log((1.0 + Math.abs(x))), x);
}

def code(x):
	return math.copysign(math.log((1.0 + math.fabs(x))), x)

function code(x)
	return copysign(log(Float64(1.0 + abs(x))), x)
end

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

code[x_] := N[With[{TMP1 = Abs[N[Log[N[(1.0 + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

\mathsf{copysign}\left(\log \left(1 + \left|x\right|\right), x\right)
Derivation

  1. Initial program 29.5%

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

  2. Taylor expanded in x around 0

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

    1. lower-+.f64N/A

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

    2. lower-fabs.f6419.3%

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

  4. Applied rewrites19.3%

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

Developer Target 1: 100.0% accurate, 0.4× speedup?

\[\begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \]
(FPCore (x)
  :precision binary64
  (let* ((t_0 (/ 1.0 (fabs x))))
  (copysign
   (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0))))
   x)))
double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return copysign(log1p((fabs(x) + (fabs(x) / (hypot(1.0, t_0) + t_0)))), x);
}

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.copySign(Math.log1p((Math.abs(x) + (Math.abs(x) / (Math.hypot(1.0, t_0) + t_0)))), x);
}

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.copysign(math.log1p((math.fabs(x) + (math.fabs(x) / (math.hypot(1.0, t_0) + t_0)))), x)

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return copysign(log1p(Float64(abs(x) + Float64(abs(x) / Float64(hypot(1.0, t_0) + t_0)))), x)
end

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + t$95$0 ^ 2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

\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}

Reproduce

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herbie shell --seed 2025313 -o setup:search
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (! :herbie-platform c (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))