?

Average Accuracy: 7.0% → 10.5%
Time: 11.7s
Precision: binary64
Cost: 103872

?

\[0 \leq x \land x \leq 0.5\]
\[\cos^{-1} \left(1 - x\right) \]
↓
\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({t_0}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + {\left(\sqrt{{\left(\sqrt[3]{t_0}\right)}^{3}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]
(FPCore (x) :precision binary64 (acos (- 1.0 x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (- (* (pow PI 3.0) 0.125) (expm1 (log1p (pow t_0 3.0))))
    (+
     (* (* PI PI) 0.25)
     (* (pow (sqrt (pow (cbrt t_0) 3.0)) 2.0) (fma PI 0.5 t_0))))))
double code(double x) {
	return acos((1.0 - x));
}
↓
double code(double x) {
	double t_0 = asin((1.0 - x));
	return ((pow(((double) M_PI), 3.0) * 0.125) - expm1(log1p(pow(t_0, 3.0)))) / (((((double) M_PI) * ((double) M_PI)) * 0.25) + (pow(sqrt(pow(cbrt(t_0), 3.0)), 2.0) * fma(((double) M_PI), 0.5, t_0)));
}

function code(x)
	return acos(Float64(1.0 - x))
end
↓
function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(Float64((pi ^ 3.0) * 0.125) - expm1(log1p((t_0 ^ 3.0)))) / Float64(Float64(Float64(pi * pi) * 0.25) + Float64((sqrt((cbrt(t_0) ^ 3.0)) ^ 2.0) * fma(pi, 0.5, t_0))))
end

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125), $MachinePrecision] - N[(Exp[N[Log[1 + N[Power[t$95$0, 3.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] + N[(N[Power[N[Sqrt[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\cos^{-1} \left(1 - x\right)
↓
\begin{array}{l}
t_0 := \sin^{-1} \left(1 - x\right)\\
\frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({t_0}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + {\left(\sqrt{{\left(\sqrt[3]{t_0}\right)}^{3}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)}
\end{array}

Error?

Target

Original7.0%
Target100.0%
Herbie10.5%
\[2 \cdot \sin^{-1} \left(\sqrt{\frac{x}{2}}\right) \]

Derivation?

  1. Initial program 7.0%

    \[\cos^{-1} \left(1 - x\right) \]

  2. Applied egg-rr7.0%

    \[\leadsto \color{blue}{\frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)}} \]
    Proof

    [Start]7.0

    \[ \cos^{-1} \left(1 - x\right) \]

    acos-asin [=>]7.0

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

    flip3-- [=>]7.0

    \[ \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)}} \]

    div-inv [=>]7.0

    \[ \frac{{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    metadata-eval [=>]7.0

    \[ \frac{{\left(\pi \cdot \color{blue}{0.5}\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    div-inv [=>]7.0

    \[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    metadata-eval [=>]7.0

    \[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    div-inv [=>]7.0

    \[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    metadata-eval [=>]7.0

    \[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    div-inv [=>]7.0

    \[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    metadata-eval [=>]7.0

    \[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot \color{blue}{0.5}\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

  3. Simplified7.0%

    \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
    Proof

    [Start]7.0

    \[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    cube-prod [=>]7.0

    \[ \frac{\color{blue}{{\pi}^{3} \cdot {0.5}^{3}} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    metadata-eval [=>]7.0

    \[ \frac{{\pi}^{3} \cdot \color{blue}{0.125} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    swap-sqr [=>]7.0

    \[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \left(0.5 \cdot 0.5\right)} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    metadata-eval [=>]7.0

    \[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \color{blue}{0.25} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    distribute-rgt-out [=>]7.0

    \[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]

    +-commutative [<=]7.0

    \[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]

    fma-def [=>]7.0

    \[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]

  4. Applied egg-rr10.5%

    \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
    Proof

    [Start]7.0

    \[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

    expm1-log1p-u [=>]10.5

    \[ \frac{{\pi}^{3} \cdot 0.125 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

  5. Applied egg-rr10.5%

    \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
    Proof

    [Start]10.5

    \[ \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

    add-sqr-sqrt [=>]10.5

    \[ \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

    pow2 [=>]10.5

    \[ \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

  6. Applied egg-rr10.5%

    \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + {\left(\sqrt{\color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
    Proof

    [Start]10.5

    \[ \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

    add-cube-cbrt [=>]10.5

    \[ \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + {\left(\sqrt{\color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

    pow3 [=>]10.5

    \[ \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + {\left(\sqrt{\color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

  7. Final simplification10.5%

    \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + {\left(\sqrt{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}\right)}^{2} \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

Alternatives

Alternative 1
Accuracy10.5%
Cost91008
\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({t_0}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \mathsf{fma}\left(\pi, 0.5, t_0\right) \cdot {\left(\sqrt{t_0}\right)}^{2}} \end{array} \]
Alternative 2
Accuracy10.5%
Cost78144
\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({t_0}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]
Alternative 3
Accuracy10.5%
Cost45440
\[\mathsf{fma}\left(\sqrt[3]{0.25 \cdot {\pi}^{2}}, {\left(\pi \cdot 0.5\right)}^{0.3333333333333333}, -\sin^{-1} \left(1 - x\right)\right) \]
Alternative 4
Accuracy9.6%
Cost32644
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi - t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\\ \end{array} \]
Alternative 5
Accuracy10.5%
Cost26048
\[\pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \]
Alternative 6
Accuracy10.5%
Cost26048
\[\pi \cdot 0.5 - {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2} \]
Alternative 7
Accuracy9.6%
Cost19844
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi - t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \end{array} \]
Alternative 8
Accuracy9.6%
Cost19716
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi - t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + t_0\right)\\ \end{array} \]
Alternative 9
Accuracy7.0%
Cost13700
\[\begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\left(1 + \pi \cdot 0.5\right) + \left(-1 - \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]
Alternative 10
Accuracy7.0%
Cost6848
\[1 + \left(\cos^{-1} \left(1 - x\right) + -1\right) \]
Alternative 11
Accuracy7.0%
Cost6848
\[-1 + \left(1 + \cos^{-1} \left(1 - x\right)\right) \]
Alternative 12
Accuracy7.0%
Cost6592
\[\cos^{-1} \left(1 - x\right) \]

Error

Reproduce?

herbie shell --seed 2023135 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))