Average Error: 0.4 → 0.4
Time: 8.6s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\begin{array}{l} t_1 := 1 - v \cdot v\\ t_2 := \pi \cdot \sqrt{2 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}\\ t_3 := t_2 \cdot t\\ t_4 := \frac{v}{t_3}\\ t_5 := \frac{v \cdot 5}{t_1}\\ \mathsf{fma}\left(1, \frac{1}{t_2 \cdot \left(t \cdot t_1\right)}, t_4 \cdot \frac{v \cdot -5}{t_1}\right) + \mathsf{fma}\left(\frac{-v}{t_3}, t_5, t_4 \cdot t_5\right) \end{array} \]
(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))
↓
(FPCore (v t)
 :precision binary64
 (let* ((t_1 (- 1.0 (* v v)))
        (t_2 (* PI (sqrt (* 2.0 (fma (* v v) -3.0 1.0)))))
        (t_3 (* t_2 t))
        (t_4 (/ v t_3))
        (t_5 (/ (* v 5.0) t_1)))
   (+
    (fma 1.0 (/ 1.0 (* t_2 (* t t_1))) (* t_4 (/ (* v -5.0) t_1)))
    (fma (/ (- v) t_3) t_5 (* t_4 t_5)))))
double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((((double) M_PI) * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
}
↓
double code(double v, double t) {
	double t_1 = 1.0 - (v * v);
	double t_2 = ((double) M_PI) * sqrt((2.0 * fma((v * v), -3.0, 1.0)));
	double t_3 = t_2 * t;
	double t_4 = v / t_3;
	double t_5 = (v * 5.0) / t_1;
	return fma(1.0, (1.0 / (t_2 * (t * t_1))), (t_4 * ((v * -5.0) / t_1))) + fma((-v / t_3), t_5, (t_4 * t_5));
}

function code(v, t)
	return Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(Float64(pi * t) * sqrt(Float64(2.0 * Float64(1.0 - Float64(3.0 * Float64(v * v)))))) * Float64(1.0 - Float64(v * v))))
end
↓
function code(v, t)
	t_1 = Float64(1.0 - Float64(v * v))
	t_2 = Float64(pi * sqrt(Float64(2.0 * fma(Float64(v * v), -3.0, 1.0))))
	t_3 = Float64(t_2 * t)
	t_4 = Float64(v / t_3)
	t_5 = Float64(Float64(v * 5.0) / t_1)
	return Float64(fma(1.0, Float64(1.0 / Float64(t_2 * Float64(t * t_1))), Float64(t_4 * Float64(Float64(v * -5.0) / t_1))) + fma(Float64(Float64(-v) / t_3), t_5, Float64(t_4 * t_5)))
end

code[v_, t_] := N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * t), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[v_, t_] := Block[{t$95$1 = N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[Sqrt[N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t), $MachinePrecision]}, Block[{t$95$4 = N[(v / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(v * 5.0), $MachinePrecision] / t$95$1), $MachinePrecision]}, N[(N[(1.0 * N[(1.0 / N[(t$95$2 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * N[(N[(v * -5.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[((-v) / t$95$3), $MachinePrecision] * t$95$5 + N[(t$95$4 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\begin{array}{l}
t_1 := 1 - v \cdot v\\
t_2 := \pi \cdot \sqrt{2 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}\\
t_3 := t_2 \cdot t\\
t_4 := \frac{v}{t_3}\\
t_5 := \frac{v \cdot 5}{t_1}\\
\mathsf{fma}\left(1, \frac{1}{t_2 \cdot \left(t \cdot t_1\right)}, t_4 \cdot \frac{v \cdot -5}{t_1}\right) + \mathsf{fma}\left(\frac{-v}{t_3}, t_5, t_4 \cdot t_5\right)
\end{array}

Error

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

  2. Applied egg-rr0.4

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

  3. Applied egg-rr0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2022210 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))