Average Error: 59.5 → 59.5
Time: 16.2s
Precision: binary64
Cost: 96960
\[0 \leq x \land x \leq 0.5\]
\[\cos^{-1} \left(1 - x\right) \]
↓
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := {t_0}^{2}\\ {\left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\sqrt[3]{t_0}}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{t_1 \cdot \sqrt[3]{t_1}}}}\right)}^{3}\right)}^{3} \end{array} \]
(FPCore (x) :precision binary64 (acos (- 1.0 x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))) (t_1 (pow t_0 2.0)))
   (pow
    (pow
     (*
      (cbrt (cbrt (cbrt (cbrt t_0))))
      (cbrt (cbrt (cbrt (* t_1 (cbrt t_1))))))
     3.0)
    3.0)))
double code(double x) {
	return acos((1.0 - x));
}
↓
double code(double x) {
	double t_0 = acos((1.0 - x));
	double t_1 = pow(t_0, 2.0);
	return pow(pow((cbrt(cbrt(cbrt(cbrt(t_0)))) * cbrt(cbrt(cbrt((t_1 * cbrt(t_1)))))), 3.0), 3.0);
}

public static double code(double x) {
	return Math.acos((1.0 - x));
}
↓
public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double t_1 = Math.pow(t_0, 2.0);
	return Math.pow(Math.pow((Math.cbrt(Math.cbrt(Math.cbrt(Math.cbrt(t_0)))) * Math.cbrt(Math.cbrt(Math.cbrt((t_1 * Math.cbrt(t_1)))))), 3.0), 3.0);
}

function code(x)
	return acos(Float64(1.0 - x))
end
↓
function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = t_0 ^ 2.0
	return (Float64(cbrt(cbrt(cbrt(cbrt(t_0)))) * cbrt(cbrt(cbrt(Float64(t_1 * cbrt(t_1)))))) ^ 3.0) ^ 3.0
end

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, N[Power[N[Power[N[(N[Power[N[Power[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[N[Power[N[(t$95$1 * N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 3.0], $MachinePrecision]]]

\cos^{-1} \left(1 - x\right)
↓
\begin{array}{l}
t_0 := \cos^{-1} \left(1 - x\right)\\
t_1 := {t_0}^{2}\\
{\left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\sqrt[3]{t_0}}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{t_1 \cdot \sqrt[3]{t_1}}}}\right)}^{3}\right)}^{3}
\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.5
Target0.0
Herbie59.5
\[2 \cdot \sin^{-1} \left(\sqrt{\frac{x}{2}}\right) \]

Derivation

  1. Initial program 59.5

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

  2. Applied egg-rr59.5

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]

  3. Applied egg-rr59.5

    \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(1 - x\right)}}\right)}^{3}\right)}}^{3} \]

  4. Applied egg-rr59.5

    \[\leadsto {\left({\color{blue}{\left({\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{3} \]

  5. Applied egg-rr59.5

    \[\leadsto {\left({\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\sqrt[3]{\cos^{-1} \left(1 - x\right)}}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{{\cos^{-1} \left(1 - x\right)}^{2} \cdot \sqrt[3]{{\cos^{-1} \left(1 - x\right)}^{2}}}}}\right)}}^{3}\right)}^{3} \]

Alternatives

Alternative 1
Error59.5
Cost90496
\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{\cos^{-1} \left(1 - x\right)}}\\ t_1 := \sqrt[3]{t_0}\\ {\left(\left({t_0}^{2} \cdot t_1\right) \cdot {t_1}^{2}\right)}^{3} \end{array} \]
Alternative 2
Error59.5
Cost77504
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ {\left({\left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\sqrt[3]{{t_0}^{3}}}}}\right)}^{3}\right)}^{2} \cdot \sqrt[3]{\sqrt[3]{t_0}}\right)}^{3} \end{array} \]
Alternative 3
Error59.5
Cost77504
\[\begin{array}{l} t_0 := \sqrt[3]{\cos^{-1} \left(1 - x\right)}\\ {\left({\left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{{t_0}^{3}}}}\right)}^{3}\right)}^{2} \cdot \sqrt[3]{t_0}\right)}^{3} \end{array} \]
Alternative 4
Error59.5
Cost64640
\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{\cos^{-1} \left(1 - x\right)}}\\ {\left({\left({\left(\sqrt[3]{t_0}\right)}^{3}\right)}^{2} \cdot t_0\right)}^{3} \end{array} \]
Alternative 5
Error59.5
Cost26436
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \ne 0:\\ \;\;\;\;\frac{\sqrt[3]{2}}{\frac{2}{t_0}} \cdot \sqrt[3]{4}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error59.5
Cost19776
\[9 \cdot {\left({\left(0.1111111111111111 \cdot \cos^{-1} \left(1 - x\right)\right)}^{3}\right)}^{0.3333333333333333} \]
Alternative 7
Error59.5
Cost13508
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \ne 0:\\ \;\;\;\;\frac{2}{\frac{2}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error59.5
Cost6848
\[27 \cdot \left(0.037037037037037035 \cdot \cos^{-1} \left(1 - x\right)\right) \]
Alternative 9
Error59.5
Cost6848
\[\left(\cos^{-1} \left(1 - x\right) \cdot 729\right) \cdot 0.0013717421124828531 \]
Alternative 10
Error59.5
Cost6592
\[\cos^{-1} \left(1 - x\right) \]

Error

Reproduce

herbie shell --seed 2023010 
(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)))