Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.0%

Time: 52.8s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\\ t_1 := -t\_0\\ \mathsf{fma}\left(\pi, 0.5, t\_0 \cdot \left(t\_0 \cdot t\_1\right)\right) + \mathsf{fma}\left(t\_1, t\_0 \cdot \sqrt[3]{\sin^{-1} \left(-1 + {v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + {v}^{2} \cdot 4\right)\right)\right)}, t\_0 \cdot {\left({\left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{2}\right)}^{3}\right) \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (cbrt (asin (/ (- 1.0 (* 5.0 (pow v 2.0))) (fma v v -1.0)))))
        (t_1 (- t_0)))
   (+
    (fma PI 0.5 (* t_0 (* t_0 t_1)))
    (fma
     t_1
     (*
      t_0
      (cbrt
       (asin
        (+
         -1.0
         (*
          (pow v 2.0)
          (+ 4.0 (* (pow v 2.0) (+ 4.0 (* (pow v 2.0) 4.0)))))))))
     (*
      t_0
      (pow
       (pow
        (cbrt (cbrt (asin (/ (+ 1.0 (* (pow v 2.0) -5.0)) (fma v v -1.0)))))
        2.0)
       3.0))))))

C

double code(double v) {
	double t_0 = cbrt(asin(((1.0 - (5.0 * pow(v, 2.0))) / fma(v, v, -1.0))));
	double t_1 = -t_0;
	return fma(((double) M_PI), 0.5, (t_0 * (t_0 * t_1))) + fma(t_1, (t_0 * cbrt(asin((-1.0 + (pow(v, 2.0) * (4.0 + (pow(v, 2.0) * (4.0 + (pow(v, 2.0) * 4.0))))))))), (t_0 * pow(pow(cbrt(cbrt(asin(((1.0 + (pow(v, 2.0) * -5.0)) / fma(v, v, -1.0))))), 2.0), 3.0)));
}

Julia

function code(v)
	t_0 = cbrt(asin(Float64(Float64(1.0 - Float64(5.0 * (v ^ 2.0))) / fma(v, v, -1.0))))
	t_1 = Float64(-t_0)
	return Float64(fma(pi, 0.5, Float64(t_0 * Float64(t_0 * t_1))) + fma(t_1, Float64(t_0 * cbrt(asin(Float64(-1.0 + Float64((v ^ 2.0) * Float64(4.0 + Float64((v ^ 2.0) * Float64(4.0 + Float64((v ^ 2.0) * 4.0))))))))), Float64(t_0 * ((cbrt(cbrt(asin(Float64(Float64(1.0 + Float64((v ^ 2.0) * -5.0)) / fma(v, v, -1.0))))) ^ 2.0) ^ 3.0))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[Power[N[ArcSin[N[(N[(1.0 - N[(5.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, N[(N[(Pi * 0.5 + N[(t$95$0 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(t$95$0 * N[Power[N[ArcSin[N[(-1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * N[(4.0 + N[(N[Power[v, 2.0], $MachinePrecision] * N[(4.0 + N[(N[Power[v, 2.0], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Power[N[Power[N[Power[N[Power[N[ArcSin[N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.9%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 98.9%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   2. div-inv 98.9%

      \[\leadsto \color{blue}{\pi \cdot \frac{1}{2}} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

   3. metadata-eval 98.9%

      \[\leadsto \pi \cdot \color{blue}{0.5} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

   4. add-cube-cbrt 96.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}} \]

   5. prod-diff 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)\right)} \]

4. Applied egg-rr 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)} \]

5. Taylor expanded in v around 0 96.6%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \color{blue}{\left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]
6. Step-by-step derivation

   1. add-cube-cbrt 97.4%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}\right) \]

   2. pow3 96.6%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}}\right) \]

   3. cbrt-prod 98.9%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot {\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}}^{3}\right) \]

7. Applied egg-rr 98.9%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{{\left({\left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{2}\right)}^{3}}\right) \]

8. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(-1 + {v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + {v}^{2} \cdot 4\right)\right)\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{2}\right)}^{3}\right) \]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ t_1 := \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\\ \mathsf{fma}\left(\pi, 0.5, t\_1 \cdot \left(-\sqrt[3]{{t\_0}^{2}}\right)\right) + \mathsf{fma}\left(-t\_1, t\_1 \cdot \sqrt[3]{\sin^{-1} \left(-1 + {v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + {v}^{2} \cdot 4\right)\right)\right)}, t\_1 \cdot {\left(\sqrt[3]{t\_0}\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (asin (/ (+ 1.0 (* (pow v 2.0) -5.0)) (fma v v -1.0))))
        (t_1 (cbrt (asin (/ (- 1.0 (* 5.0 (pow v 2.0))) (fma v v -1.0))))))
   (+
    (fma PI 0.5 (* t_1 (- (cbrt (pow t_0 2.0)))))
    (fma
     (- t_1)
     (*
      t_1
      (cbrt
       (asin
        (+
         -1.0
         (*
          (pow v 2.0)
          (+ 4.0 (* (pow v 2.0) (+ 4.0 (* (pow v 2.0) 4.0)))))))))
     (* t_1 (pow (cbrt t_0) 2.0))))))

C

double code(double v) {
	double t_0 = asin(((1.0 + (pow(v, 2.0) * -5.0)) / fma(v, v, -1.0)));
	double t_1 = cbrt(asin(((1.0 - (5.0 * pow(v, 2.0))) / fma(v, v, -1.0))));
	return fma(((double) M_PI), 0.5, (t_1 * -cbrt(pow(t_0, 2.0)))) + fma(-t_1, (t_1 * cbrt(asin((-1.0 + (pow(v, 2.0) * (4.0 + (pow(v, 2.0) * (4.0 + (pow(v, 2.0) * 4.0))))))))), (t_1 * pow(cbrt(t_0), 2.0)));
}

Julia

function code(v)
	t_0 = asin(Float64(Float64(1.0 + Float64((v ^ 2.0) * -5.0)) / fma(v, v, -1.0)))
	t_1 = cbrt(asin(Float64(Float64(1.0 - Float64(5.0 * (v ^ 2.0))) / fma(v, v, -1.0))))
	return Float64(fma(pi, 0.5, Float64(t_1 * Float64(-cbrt((t_0 ^ 2.0))))) + fma(Float64(-t_1), Float64(t_1 * cbrt(asin(Float64(-1.0 + Float64((v ^ 2.0) * Float64(4.0 + Float64((v ^ 2.0) * Float64(4.0 + Float64((v ^ 2.0) * 4.0))))))))), Float64(t_1 * (cbrt(t_0) ^ 2.0))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[ArcSin[N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[ArcSin[N[(N[(1.0 - N[(5.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(Pi * 0.5 + N[(t$95$1 * (-N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + N[((-t$95$1) * N[(t$95$1 * N[Power[N[ArcSin[N[(-1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * N[(4.0 + N[(N[Power[v, 2.0], $MachinePrecision] * N[(4.0 + N[(N[Power[v, 2.0], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.9%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 98.9%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   2. div-inv 98.9%

      \[\leadsto \color{blue}{\pi \cdot \frac{1}{2}} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

   3. metadata-eval 98.9%

      \[\leadsto \pi \cdot \color{blue}{0.5} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

   4. add-cube-cbrt 96.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}} \]

   5. prod-diff 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)\right)} \]

4. Applied egg-rr 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)} \]

5. Taylor expanded in v around 0 96.6%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \color{blue}{\left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]
6. Step-by-step derivation

   1. cbrt-unprod 98.9%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

   2. rem-3cbrt-rft 98.9%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

   3. rem-3cbrt-rft 97.4%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \cdot \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

   4. pow2 97.4%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)}^{2}}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

7. Applied egg-rr 98.9%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{\sqrt[3]{{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]
8. Step-by-step derivation

   1. sqr-neg 98.9%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{\left(\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right) \cdot \left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)}\right) \]

   2. pow2 98.9%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{{\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}}\right) \]

9. Applied egg-rr 98.9%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}}\right) \]

10. Final simplification 98.9%

    \[\leadsto \mathsf{fma}\left(\pi, 0.5, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(-\sqrt[3]{{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(-1 + {v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + {v}^{2} \cdot 4\right)\right)\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}\right) \]
11. Add Preprocessing

Alternative 3: 99.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 5 \cdot {v}^{2}\\ t_1 := \sin^{-1} \left(\frac{t\_0}{{v}^{2} + -1}\right)\\ \mathsf{fma}\left(\pi, 0.5, \sqrt[3]{\sin^{-1} \left(\frac{t\_0}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(-\sqrt[3]{{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}}\right)\right) + \left(t\_1 - \sqrt[3]{\sin^{-1} \left(-1 + \left({v}^{2} \cdot 4 + \left(4 \cdot {v}^{4} + 4 \cdot {v}^{6}\right)\right)\right) \cdot {t\_1}^{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 5.0 (pow v 2.0))))
        (t_1 (asin (/ t_0 (+ (pow v 2.0) -1.0)))))
   (+
    (fma
     PI
     0.5
     (*
      (cbrt (asin (/ t_0 (fma v v -1.0))))
      (-
       (cbrt
        (pow (asin (/ (+ 1.0 (* (pow v 2.0) -5.0)) (fma v v -1.0))) 2.0)))))
    (-
     t_1
     (cbrt
      (*
       (asin
        (+
         -1.0
         (+ (* (pow v 2.0) 4.0) (+ (* 4.0 (pow v 4.0)) (* 4.0 (pow v 6.0))))))
       (pow t_1 2.0)))))))

C

double code(double v) {
	double t_0 = 1.0 - (5.0 * pow(v, 2.0));
	double t_1 = asin((t_0 / (pow(v, 2.0) + -1.0)));
	return fma(((double) M_PI), 0.5, (cbrt(asin((t_0 / fma(v, v, -1.0)))) * -cbrt(pow(asin(((1.0 + (pow(v, 2.0) * -5.0)) / fma(v, v, -1.0))), 2.0)))) + (t_1 - cbrt((asin((-1.0 + ((pow(v, 2.0) * 4.0) + ((4.0 * pow(v, 4.0)) + (4.0 * pow(v, 6.0)))))) * pow(t_1, 2.0))));
}

Julia

function code(v)
	t_0 = Float64(1.0 - Float64(5.0 * (v ^ 2.0)))
	t_1 = asin(Float64(t_0 / Float64((v ^ 2.0) + -1.0)))
	return Float64(fma(pi, 0.5, Float64(cbrt(asin(Float64(t_0 / fma(v, v, -1.0)))) * Float64(-cbrt((asin(Float64(Float64(1.0 + Float64((v ^ 2.0) * -5.0)) / fma(v, v, -1.0))) ^ 2.0))))) + Float64(t_1 - cbrt(Float64(asin(Float64(-1.0 + Float64(Float64((v ^ 2.0) * 4.0) + Float64(Float64(4.0 * (v ^ 4.0)) + Float64(4.0 * (v ^ 6.0)))))) * (t_1 ^ 2.0)))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(1.0 - N[(5.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(t$95$0 / N[(N[Power[v, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(Pi * 0.5 + N[(N[Power[N[ArcSin[N[(t$95$0 / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] * (-N[Power[N[Power[N[ArcSin[N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Power[N[(N[ArcSin[N[(-1.0 + N[(N[(N[Power[v, 2.0], $MachinePrecision] * 4.0), $MachinePrecision] + N[(N[(4.0 * N[Power[v, 4.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[Power[v, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.9%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 98.9%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   2. div-inv 98.9%

      \[\leadsto \color{blue}{\pi \cdot \frac{1}{2}} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

   3. metadata-eval 98.9%

      \[\leadsto \pi \cdot \color{blue}{0.5} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

   4. add-cube-cbrt 96.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}} \]

   5. prod-diff 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)\right)} \]

4. Applied egg-rr 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)} \]

5. Taylor expanded in v around 0 96.6%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \color{blue}{\left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]
6. Step-by-step derivation

   1. cbrt-unprod 98.9%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

   2. rem-3cbrt-rft 98.9%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

   3. rem-3cbrt-rft 97.4%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \cdot \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

   4. pow2 97.4%

      \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)}^{2}}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

7. Applied egg-rr 98.9%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \color{blue}{\sqrt[3]{{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}}}\right) + \mathsf{fma}\left(-\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

8. Taylor expanded in v around inf 98.9%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}}\right) + \color{blue}{\left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right) + -1 \cdot \sqrt[3]{\sin^{-1} \left(\left(4 \cdot {v}^{2} + \left(4 \cdot {v}^{4} + 4 \cdot {v}^{6}\right)\right) - 1\right) \cdot {\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{2}}\right)} \]

9. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(\pi, 0.5, \sqrt[3]{\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(-\sqrt[3]{{\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}}\right)\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} + -1}\right) - \sqrt[3]{\sin^{-1} \left(-1 + \left({v}^{2} \cdot 4 + \left(4 \cdot {v}^{4} + 4 \cdot {v}^{6}\right)\right)\right) \cdot {\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} + -1}\right)}^{2}}\right) \]
10. Add Preprocessing

Alternative 4: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot 9} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (exp
  (*
   (log (cbrt (cbrt (acos (/ (fma (pow v 2.0) -5.0 1.0) (fma v v -1.0))))))
   9.0)))

C

double code(double v) {
	return exp((log(cbrt(cbrt(acos((fma(pow(v, 2.0), -5.0, 1.0) / fma(v, v, -1.0)))))) * 9.0));
}

Julia

function code(v)
	return exp(Float64(log(cbrt(cbrt(acos(Float64(fma((v ^ 2.0), -5.0, 1.0) / fma(v, v, -1.0)))))) * 9.0))
end

Wolfram

code[v_] := N[Exp[N[(N[Log[N[Power[N[Power[N[ArcCos[N[(N[(N[Power[v, 2.0], $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * 9.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 96.6%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}} \]

   2. pow3 97.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)}^{3}} \]

   3. pow2 97.4%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{{v}^{2}}}{v \cdot v - 1}\right)}\right)}^{3} \]

   4. fma-neg 97.4%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)}\right)}^{3} \]

   5. metadata-eval 97.4%

      \[\leadsto {\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right)}\right)}^{3} \]

4. Applied egg-rr 97.4%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{3}} \]
5. Step-by-step derivation

   1. add-cube-cbrt 96.6%

      \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}}^{3} \]

   2. pow3 96.6%

      \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}}^{3} \]

   3. sub-neg 96.6%

      \[\leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot {v}^{2}\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}^{3} \]

   4. *-commutative 96.6%

      \[\leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 + \left(-\color{blue}{{v}^{2} \cdot 5}\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}^{3} \]

   5. distribute-rgt-neg-in 96.6%

      \[\leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 + \color{blue}{{v}^{2} \cdot \left(-5\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}^{3} \]

   6. metadata-eval 96.6%

      \[\leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 + {v}^{2} \cdot \color{blue}{-5}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}^{3} \]

6. Applied egg-rr 96.6%

   \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{3}\right)}}^{3} \]
7. Step-by-step derivation

   1. pow-pow 97.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{\left(3 \cdot 3\right)}} \]

   2. pow-to-exp 98.9%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot \left(3 \cdot 3\right)}} \]

   3. +-commutative 98.9%

      \[\leadsto e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\color{blue}{{v}^{2} \cdot -5 + 1}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot \left(3 \cdot 3\right)} \]

   4. fma-define 98.9%

      \[\leadsto e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left({v}^{2}, -5, 1\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot \left(3 \cdot 3\right)} \]

   5. metadata-eval 98.9%

      \[\leadsto e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot \color{blue}{9}} \]

8. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot 9}} \]

9. Final simplification 98.9%

   \[\leadsto e^{\log \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right) \cdot 9} \]
10. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{-1 + v \cdot v}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (+ -1.0 (* v v)))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / (-1.0 + (v * v))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((-1.0d0) + (v * v))))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / (-1.0 + (v * v))));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / (-1.0 + (v * v))))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(-1.0 + Float64(v * v))))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / (-1.0 + (v * v))));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Final simplification 98.9%

   \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{-1 + v \cdot v}\right) \]
4. Add Preprocessing

Alternative 6: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(-1 + 4 \cdot \left(v \cdot v\right)\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos (+ -1.0 (* 4.0 (* v v)))))

C

double code(double v) {
	return acos((-1.0 + (4.0 * (v * v))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((-1.0d0) + (4.0d0 * (v * v))))
end function

Java

public static double code(double v) {
	return Math.acos((-1.0 + (4.0 * (v * v))));
}

Python

def code(v):
	return math.acos((-1.0 + (4.0 * (v * v))))

Julia

function code(v)
	return acos(Float64(-1.0 + Float64(4.0 * Float64(v * v))))
end

MATLAB

function tmp = code(v)
	tmp = acos((-1.0 + (4.0 * (v * v))));
end

Wolfram

code[v_] := N[ArcCos[N[(-1.0 + N[(4.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0 98.5%

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)} \]
4. Step-by-step derivation

   1. unpow2 98.5%

      \[\leadsto \cos^{-1} \left(4 \cdot \color{blue}{\left(v \cdot v\right)} - 1\right) \]

5. Applied egg-rr 98.5%

   \[\leadsto \cos^{-1} \left(4 \cdot \color{blue}{\left(v \cdot v\right)} - 1\right) \]

6. Final simplification 98.5%

   \[\leadsto \cos^{-1} \left(-1 + 4 \cdot \left(v \cdot v\right)\right) \]
7. Add Preprocessing

Alternative 7: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -1 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos -1.0))

C

double code(double v) {
	return acos(-1.0);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos((-1.0d0))
end function

Java

public static double code(double v) {
	return Math.acos(-1.0);
}

Python

def code(v):
	return math.acos(-1.0)

Julia

function code(v)
	return acos(-1.0)
end

MATLAB

function tmp = code(v)
	tmp = acos(-1.0);
end

Wolfram

code[v_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0 97.7%

   \[\leadsto \cos^{-1} \color{blue}{-1} \]

4. Final simplification 97.7%

   \[\leadsto \cos^{-1} -1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024082 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))