Average Error: 0.5 → 0.5
Time: 7.6s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[\begin{array}{l} t_0 := \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \frac{0.125 \cdot {\pi}^{3} - {t_0}^{3}}{0.5 \cdot \left(\pi \cdot t_0\right) + \left(0.25 \cdot {\pi}^{2} + {t_0}^{2}\right)} \end{array} \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (let* ((t_0 (asin (/ (fma v (* v -5.0) 1.0) (fma v v -1.0)))))
   (/
    (- (* 0.125 (pow PI 3.0)) (pow t_0 3.0))
    (+ (* 0.5 (* PI t_0)) (+ (* 0.25 (pow PI 2.0)) (pow t_0 2.0))))))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	double t_0 = asin((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0)));
	return ((0.125 * pow(((double) M_PI), 3.0)) - pow(t_0, 3.0)) / ((0.5 * (((double) M_PI) * t_0)) + ((0.25 * pow(((double) M_PI), 2.0)) + pow(t_0, 2.0)));
}

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end
↓
function code(v)
	t_0 = asin(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0)))
	return Float64(Float64(Float64(0.125 * (pi ^ 3.0)) - (t_0 ^ 3.0)) / Float64(Float64(0.5 * Float64(pi * t_0)) + Float64(Float64(0.25 * (pi ^ 2.0)) + (t_0 ^ 2.0))))
end

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]
↓
code[v_] := Block[{t$95$0 = N[ArcSin[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(0.125 * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\begin{array}{l}
t_0 := \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\
\frac{0.125 \cdot {\pi}^{3} - {t_0}^{3}}{0.5 \cdot \left(\pi \cdot t_0\right) + \left(0.25 \cdot {\pi}^{2} + {t_0}^{2}\right)}
\end{array}

Error

Bits error versus v

Derivation

  1. Initial program 0.5

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

  2. Simplified0.5

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

  3. Applied egg-rr0.5

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

  4. Taylor expanded in v around 0 0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2022178 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))