Rust f32::asinh

Percentage Accurate: 38.6% → 99.5%
Time: 2.7s
Alternatives: 5
Speedup: 1.6×

Specification

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

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 5 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: 38.6% 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, 1.6× speedup?

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

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

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

\begin{array}{l}

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

  1. Initial program 38.6%

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

  3. Applied rewrites99.5%

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

Alternative 2: 45.1% accurate, 0.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary32
 (if (<=
      (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)
      0.10000000149011612)
   (copysign (log (+ (fabs x) 1.0)) x)
   (copysign (log (+ x (fabs x))) x)))
float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= 0.10000000149011612f) {
		tmp = copysignf(logf((fabsf(x) + 1.0f)), x);
	} else {
		tmp = copysignf(logf((x + fabsf(x))), x);
	}
	return tmp;
}

function code(x)
	tmp = Float32(0.0)
	if (copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x) <= Float32(0.10000000149011612))
		tmp = copysign(log(Float32(abs(x) + Float32(1.0))), x);
	else
		tmp = copysign(log(Float32(x + abs(x))), x);
	end
	return tmp
end

function tmp_2 = code(x)
	tmp = single(0.0);
	if ((sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))))) <= single(0.10000000149011612))
		tmp = sign(x) * abs(log((abs(x) + single(1.0))));
	else
		tmp = sign(x) * abs(log((x + abs(x))));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.10000000149011612:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + 1\right), x\right)\\

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


\end{array}
\end{array}
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) < 0.100000001

    1. Initial program 38.6%

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

    2. Taylor expanded in x around 0

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

    4. Applied rewrites31.7%

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

    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 38.6%

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

    2. Taylor expanded in x around inf

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

    4. Applied rewrites30.0%

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

    5. Taylor expanded in x around 0

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

    7. Applied rewrites30.0%

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

Alternative 3: 30.0% accurate, 1.6× speedup?

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

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

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

\begin{array}{l}

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

  1. Initial program 38.6%

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

  2. Taylor expanded in x around inf

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

  4. Applied rewrites30.0%

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

  5. Taylor expanded in x around 0

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

  7. Applied rewrites30.0%

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

Alternative 4: 27.5% accurate, 2.0× speedup?

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

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

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

\begin{array}{l}

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

  1. Initial program 38.6%

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

  2. Taylor expanded in x around inf

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

  4. Applied rewrites30.0%

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

  5. Taylor expanded in x around 0

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

  7. Applied rewrites30.0%

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

  8. Taylor expanded in x around 0

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

  10. Applied rewrites27.5%

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

Alternative 5: 13.6% accurate, 2.0× speedup?

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

function code(x)
	return copysign(log(Float32(-x)), x)
end

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

\begin{array}{l}

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

  1. Initial program 38.6%

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

  2. Taylor expanded in x around -inf

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

  4. Applied rewrites13.6%

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

  6. Applied rewrites13.6%

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

Developer Target 1: 99.6% accurate, 0.4× 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 2025161 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :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))