Ian Simplification

Percentage Accurate: 7.1% → 8.6%
Time: 6.9s
Alternatives: 2
Speedup: 1.1×

Specification

?
\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))
double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\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 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: 7.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))
double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)
\end{array}

Alternative 1: 8.6% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), -4, \pi\right) \cdot -0.5 \end{array} \]
(FPCore (x)
 :precision binary64
 (* (fma (acos (sqrt (fma x -0.5 0.5))) -4.0 PI) -0.5))
double code(double x) {
	return fma(acos(sqrt(fma(x, -0.5, 0.5))), -4.0, ((double) M_PI)) * -0.5;
}

function code(x)
	return Float64(fma(acos(sqrt(fma(x, -0.5, 0.5))), -4.0, pi) * -0.5)
end

code[x_] := N[(N[(N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * -4.0 + Pi), $MachinePrecision] * -0.5), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), -4, \pi\right) \cdot -0.5
\end{array}
Derivation

  1. Initial program 7.1%

    \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

  3. Applied rewrites8.6%

    \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot 2\right)} \]

  5. Applied rewrites8.6%

    \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot 2 - 0.5 \cdot \pi} \]
  6. Step-by-step derivation

    1. lift--.f64N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2 - \frac{1}{2} \cdot \pi} \]

    2. lift-*.f64N/A

      \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2 - \color{blue}{\frac{1}{2} \cdot \pi} \]

    3. *-commutativeN/A

      \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2 - \color{blue}{\pi \cdot \frac{1}{2}} \]

    4. metadata-evalN/A

      \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2 - \pi \cdot \color{blue}{\frac{1}{2}} \]

    5. mult-flipN/A

      \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2 - \color{blue}{\frac{\pi}{2}} \]

    6. frac-2negN/A

      \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2 - \color{blue}{\frac{\mathsf{neg}\left(\pi\right)}{\mathsf{neg}\left(2\right)}} \]

    7. metadata-evalN/A

      \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2 - \frac{\mathsf{neg}\left(\pi\right)}{\color{blue}{-2}} \]

    8. sub-to-fractionN/A

      \[\leadsto \color{blue}{\frac{\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2\right) \cdot -2 - \left(\mathsf{neg}\left(\pi\right)\right)}{-2}} \]

    9. mult-flipN/A

      \[\leadsto \color{blue}{\left(\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2\right) \cdot -2 - \left(\mathsf{neg}\left(\pi\right)\right)\right) \cdot \frac{1}{-2}} \]

    10. metadata-evalN/A

      \[\leadsto \left(\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2\right) \cdot -2 - \left(\mathsf{neg}\left(\pi\right)\right)\right) \cdot \color{blue}{\frac{-1}{2}} \]

    11. lower-*.f64N/A

      \[\leadsto \color{blue}{\left(\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right) \cdot 2\right) \cdot -2 - \left(\mathsf{neg}\left(\pi\right)\right)\right) \cdot \frac{-1}{2}} \]

  7. Applied rewrites8.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), -4, \pi\right) \cdot -0.5} \]
  8. Add Preprocessing

Alternative 2: 5.4% accurate, 1.5× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos^{-1} \left(\sqrt{0.5}\right), 2, \pi \cdot -0.5\right) \end{array} \]
(FPCore (x) :precision binary64 (fma (acos (sqrt 0.5)) 2.0 (* PI -0.5)))
double code(double x) {
	return fma(acos(sqrt(0.5)), 2.0, (((double) M_PI) * -0.5));
}

function code(x)
	return fma(acos(sqrt(0.5)), 2.0, Float64(pi * -0.5))
end

code[x_] := N[(N[ArcCos[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * 2.0 + N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(\cos^{-1} \left(\sqrt{0.5}\right), 2, \pi \cdot -0.5\right)
\end{array}
Derivation

  1. Initial program 7.1%

    \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

  3. Applied rewrites8.6%

    \[\leadsto \frac{\pi}{2} - \color{blue}{\left(\pi + \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot 2\right)} \]

  5. Applied rewrites8.6%

    \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot 2 - 0.5 \cdot \pi} \]

  6. Taylor expanded in x around 0

    \[\leadsto \cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2}}}\right) \cdot 2 - \frac{1}{2} \cdot \pi \]
  7. Step-by-step derivation

    1. Applied rewrites5.4%

      \[\leadsto \cos^{-1} \left(\sqrt{\color{blue}{0.5}}\right) \cdot 2 - 0.5 \cdot \pi \]

    3. Applied rewrites5.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{0.5}\right), 2, \pi \cdot -0.5\right)} \]
    4. Add Preprocessing

    Reproduce

    ?
    herbie shell --seed 2025149 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))