Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 38.4s

Alternatives: 6

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: 99.1% 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: 99.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ t_1 := \sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\\ t_2 := {t_1}^{2}\\ t_3 := t_1 \cdot t_2\\ \frac{{\pi}^{3} \cdot 0.125 - {t_0}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \left(\mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \sqrt[3]{\pi}, -t_3\right) + \mathsf{fma}\left(-t_1, t_2, t_3\right)\right)} \end{array} \end{array} \]

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 (acos (/ (fma (pow v 2.0) -5.0 1.0) (fma v v -1.0)))))
        (t_2 (pow t_1 2.0))
        (t_3 (* t_1 t_2)))
   (/
    (- (* (pow PI 3.0) 0.125) (pow t_0 3.0))
    (+
     (* (* PI PI) 0.25)
     (*
      t_0
      (+
       (fma (pow (cbrt PI) 2.0) (cbrt PI) (- t_3))
       (fma (- t_1) t_2 t_3)))))))

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(acos((fma(pow(v, 2.0), -5.0, 1.0) / fma(v, v, -1.0))));
	double t_2 = pow(t_1, 2.0);
	double t_3 = t_1 * t_2;
	return ((pow(((double) M_PI), 3.0) * 0.125) - pow(t_0, 3.0)) / (((((double) M_PI) * ((double) M_PI)) * 0.25) + (t_0 * (fma(pow(cbrt(((double) M_PI)), 2.0), cbrt(((double) M_PI)), -t_3) + fma(-t_1, t_2, t_3))));
}

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(acos(Float64(fma((v ^ 2.0), -5.0, 1.0) / fma(v, v, -1.0))))
	t_2 = t_1 ^ 2.0
	t_3 = Float64(t_1 * t_2)
	return Float64(Float64(Float64((pi ^ 3.0) * 0.125) - (t_0 ^ 3.0)) / Float64(Float64(Float64(pi * pi) * 0.25) + Float64(t_0 * Float64(fma((cbrt(pi) ^ 2.0), cbrt(pi), Float64(-t_3)) + fma(Float64(-t_1), t_2, t_3)))))
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[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]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, N[(N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] + N[(t$95$0 * N[(N[(N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision] + (-t$95$3)), $MachinePrecision] + N[((-t$95$1) * t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 99.3%

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

   1. acos-asin 99.3%

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

   2. flip3-- 99.4%

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

3. Applied egg-rr 99.4%

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

   1. cube-prod 99.4%

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

   2. metadata-eval 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot \color{blue}{0.125} - {\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   3. cancel-sign-sub-inv 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{\color{blue}{1 + \left(-5\right) \cdot {v}^{2}}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   4. metadata-eval 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + \color{blue}{-5} \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   5. *-commutative 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + \color{blue}{{v}^{2} \cdot -5}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]
6. Step-by-step derivation

   1. fma-udef 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]

   2. add-sqr-sqrt 97.9%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot 0.5 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   3. sqrt-prod 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\sqrt{\pi \cdot \pi}} \cdot 0.5 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   4. metadata-eval 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\pi \cdot \pi} \cdot \color{blue}{\sqrt{0.25}} + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   5. sqrt-prod 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25}} + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   6. asin-acos 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)} \]

   7. div-inv 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   8. metadata-eval 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   9. add-sqr-sqrt 97.9%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot 0.5 - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   10. sqrt-prod 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\color{blue}{\sqrt{\pi \cdot \pi}} \cdot 0.5 - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   11. metadata-eval 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\sqrt{\pi \cdot \pi} \cdot \color{blue}{\sqrt{0.25}} - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   12. sqrt-prod 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\color{blue}{\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25}} - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   13. *-commutative 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} - \cos^{-1} \left(\frac{1 + \color{blue}{-5 \cdot {v}^{2}}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   14. metadata-eval 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} - \cos^{-1} \left(\frac{1 + \color{blue}{\left(-5\right)} \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   15. cancel-sign-sub-inv 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} - \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot {v}^{2}}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \color{blue}{\left(\left(\pi \cdot 0.5 + \pi \cdot 0.5\right) - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]
8. Step-by-step derivation

   1. distribute-lft-out 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\pi \cdot \left(0.5 + 0.5\right)} - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   2. metadata-eval 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\pi \cdot \color{blue}{1} - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   3. *-rgt-identity 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\pi} - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

9. Simplified 99.4%

   \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \color{blue}{\left(\pi - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]
10. Step-by-step derivation

    1. add-cube-cbrt 95.9%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}} - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

    2. add-cube-cbrt 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi} - \color{blue}{\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)} \]

    3. prod-diff 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}, \sqrt[3]{\pi}, -\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}, \sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right)\right)}} \]

11. Applied egg-rr 99.4%

    \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \color{blue}{\left(\mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \sqrt[3]{\pi}, -\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot {\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}, {\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}, \sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot {\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}\right)\right)}} \]

12. Final simplification 99.4%

    \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \sqrt[3]{\pi}, -\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot {\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}, {\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}, \sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot {\left(\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{2}\right)\right)} \]

Alternative 2: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \frac{{\pi}^{3} \cdot 0.125 - {t_0}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \left(\pi - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (asin (/ (+ 1.0 (* (pow v 2.0) -5.0)) (fma v v -1.0)))))
   (/
    (- (* (pow PI 3.0) 0.125) (pow t_0 3.0))
    (+
     (* (* PI PI) 0.25)
     (* t_0 (- PI (acos (/ (fma (pow v 2.0) -5.0 1.0) (fma v v -1.0)))))))))

C

double code(double v) {
	double t_0 = asin(((1.0 + (pow(v, 2.0) * -5.0)) / fma(v, v, -1.0)));
	return ((pow(((double) M_PI), 3.0) * 0.125) - pow(t_0, 3.0)) / (((((double) M_PI) * ((double) M_PI)) * 0.25) + (t_0 * (((double) M_PI) - acos((fma(pow(v, 2.0), -5.0, 1.0) / fma(v, v, -1.0))))));
}

Julia

function code(v)
	t_0 = asin(Float64(Float64(1.0 + Float64((v ^ 2.0) * -5.0)) / fma(v, v, -1.0)))
	return Float64(Float64(Float64((pi ^ 3.0) * 0.125) - (t_0 ^ 3.0)) / Float64(Float64(Float64(pi * pi) * 0.25) + Float64(t_0 * Float64(pi - acos(Float64(fma((v ^ 2.0), -5.0, 1.0) / fma(v, v, -1.0)))))))
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]}, N[(N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] + N[(t$95$0 * N[(Pi - N[ArcCos[N[(N[(N[Power[v, 2.0], $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

   1. acos-asin 99.3%

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

   2. flip3-- 99.4%

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

3. Applied egg-rr 99.4%

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

   1. cube-prod 99.4%

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

   2. metadata-eval 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot \color{blue}{0.125} - {\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   3. cancel-sign-sub-inv 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{\color{blue}{1 + \left(-5\right) \cdot {v}^{2}}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   4. metadata-eval 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + \color{blue}{-5} \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   5. *-commutative 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + \color{blue}{{v}^{2} \cdot -5}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]
6. Step-by-step derivation

   1. fma-udef 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]

   2. add-sqr-sqrt 97.9%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot 0.5 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   3. sqrt-prod 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\sqrt{\pi \cdot \pi}} \cdot 0.5 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   4. metadata-eval 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\pi \cdot \pi} \cdot \color{blue}{\sqrt{0.25}} + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   5. sqrt-prod 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25}} + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   6. asin-acos 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)} \]

   7. div-inv 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   8. metadata-eval 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   9. add-sqr-sqrt 97.9%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot 0.5 - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   10. sqrt-prod 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\color{blue}{\sqrt{\pi \cdot \pi}} \cdot 0.5 - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   11. metadata-eval 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\sqrt{\pi \cdot \pi} \cdot \color{blue}{\sqrt{0.25}} - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   12. sqrt-prod 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\color{blue}{\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25}} - \cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   13. *-commutative 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} - \cos^{-1} \left(\frac{1 + \color{blue}{-5 \cdot {v}^{2}}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   14. metadata-eval 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} - \cos^{-1} \left(\frac{1 + \color{blue}{\left(-5\right)} \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   15. cancel-sign-sub-inv 99.4%

       \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} + \left(\sqrt{\left(\pi \cdot \pi\right) \cdot 0.25} - \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot {v}^{2}}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \color{blue}{\left(\left(\pi \cdot 0.5 + \pi \cdot 0.5\right) - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]
8. Step-by-step derivation

   1. distribute-lft-out 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\pi \cdot \left(0.5 + 0.5\right)} - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   2. metadata-eval 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\pi \cdot \color{blue}{1} - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

   3. *-rgt-identity 99.4%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\color{blue}{\pi} - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

9. Simplified 99.4%

   \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \color{blue}{\left(\pi - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}} \]

10. Final simplification 99.4%

    \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\pi - \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \]

Alternative 3: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\log \cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\\ e^{t_0 \cdot {\left({\left(\sqrt[3]{t_0}\right)}^{2}\right)}^{3}} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0
         (cbrt (log (acos (/ (fma (pow v 2.0) -5.0 1.0) (fma v v -1.0)))))))
   (exp (* t_0 (pow (pow (cbrt t_0) 2.0) 3.0)))))

C

double code(double v) {
	double t_0 = cbrt(log(acos((fma(pow(v, 2.0), -5.0, 1.0) / fma(v, v, -1.0)))));
	return exp((t_0 * pow(pow(cbrt(t_0), 2.0), 3.0)));
}

Julia

function code(v)
	t_0 = cbrt(log(acos(Float64(fma((v ^ 2.0), -5.0, 1.0) / fma(v, v, -1.0)))))
	return exp(Float64(t_0 * ((cbrt(t_0) ^ 2.0) ^ 3.0)))
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. add-exp-log 99.3%

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

   2. pow2 99.3%

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

   3. fma-neg 99.3%

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

   4. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. add-cube-cbrt 95.5%

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

   2. pow3 95.5%

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

   3. cancel-sign-sub-inv 95.5%

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

   4. metadata-eval 95.5%

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

   5. *-commutative 95.5%

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

   6. +-commutative 95.5%

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

   7. fma-def 95.5%

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

5. Applied egg-rr 95.5%

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

   1. add-cube-cbrt 99.4%

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

   2. unpow-prod-down 97.9%

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

   3. pow2 97.9%

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

   4. pow3 97.9%

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

   5. add-cube-cbrt 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 99.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. add-exp-log 99.3%

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

   2. pow2 99.3%

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

   3. fma-neg 99.3%

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

   4. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. add-log-exp 99.3%

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

   2. exp-prod 99.3%

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

5. Applied egg-rr 99.3%

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

6. Final simplification 99.3%

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

Alternative 5: 99.1% 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 99.3%

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

2. Final simplification 99.3%

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

Alternative 6: 98.0% 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 99.3%

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

2. Taylor expanded in v around 0 98.1%

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

3. Final simplification 98.1%

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

Reproduce

herbie shell --seed 2023309 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))