Average Error: 0.6 → 0.6
Time: 7.5s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[\begin{array}{l} t_0 := v \cdot \left(v \cdot -5\right)\\ \cos^{-1} \left(\frac{\frac{t_0 \cdot t_0 + -1}{t_0 + -1}}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (let* ((t_0 (* v (* v -5.0))))
   (acos (/ (/ (+ (* t_0 t_0) -1.0) (+ t_0 -1.0)) (fma v v -1.0)))))
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	double t_0 = v * (v * -5.0);
	return acos(((((t_0 * t_0) + -1.0) / (t_0 + -1.0)) / fma(v, v, -1.0)));
}

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 = Float64(v * Float64(v * -5.0))
	return acos(Float64(Float64(Float64(Float64(t_0 * t_0) + -1.0) / Float64(t_0 + -1.0)) / fma(v, v, -1.0)))
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[(v * N[(v * -5.0), $MachinePrecision]), $MachinePrecision]}, N[ArcCos[N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $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 := v \cdot \left(v \cdot -5\right)\\
\cos^{-1} \left(\frac{\frac{t_0 \cdot t_0 + -1}{t_0 + -1}}{\mathsf{fma}\left(v, v, -1\right)}\right)
\end{array}

Error

Derivation

  1. Initial program 0.6

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

  2. Simplified0.6

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

  3. Applied egg-rr0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2022209 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))