Average Error: 0.4 → 0.3
Time: 43.8s
Precision: binary64
Cost: 27008
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\frac{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi}}{t}}{\sqrt{2 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}}}{1 - v \cdot v} \]
(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
 (/
  (/ (/ (/ (fma v (* v -5.0) 1.0) PI) t) (sqrt (* 2.0 (fma (* v v) -3.0 1.0))))
  (- 1.0 (* v v))))
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) {
	return (((fma(v, (v * -5.0), 1.0) / ((double) M_PI)) / t) / sqrt((2.0 * fma((v * v), -3.0, 1.0)))) / (1.0 - (v * v));
}

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)
	return Float64(Float64(Float64(Float64(fma(v, Float64(v * -5.0), 1.0) / pi) / t) / sqrt(Float64(2.0 * fma(Float64(v * v), -3.0, 1.0)))) / Float64(1.0 - Float64(v * v)))
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_] := N[(N[(N[(N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / Pi), $MachinePrecision] / t), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(v * v), $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)}
↓
\frac{\frac{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi}}{t}}{\sqrt{2 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}}}{1 - v \cdot v}

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

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

  3. Simplified0.3

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

  4. Applied egg-rr0.3

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

  5. Applied egg-rr0.3

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

Alternatives

Alternative 1
Error0.4
Cost20800
\[\frac{\frac{\frac{\frac{-1 - v \cdot \left(v \cdot -5\right)}{t}}{-\pi}}{\sqrt{2 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}}}{1 - v \cdot v} \]
Alternative 2
Error0.3
Cost20736
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\sqrt{2 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} \cdot \pi\right) \cdot t\right) \cdot \left(1 - v \cdot v\right)} \]
Alternative 3
Error0.3
Cost14464
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(t \cdot \left(\sqrt{2 \cdot \left(1 - \left(3 \cdot v\right) \cdot v\right)} \cdot \pi\right)\right) \cdot \left(1 - v \cdot v\right)} \]
Alternative 4
Error1.0
Cost13184
\[\frac{\frac{1}{\pi}}{\sqrt{2} \cdot t} \]
Alternative 5
Error1.1
Cost13184
\[\frac{\frac{\frac{1}{t}}{\pi}}{\sqrt{2}} \]
Alternative 6
Error1.1
Cost13184
\[\frac{\frac{\frac{1}{t}}{\sqrt{2}}}{\pi} \]
Alternative 7
Error1.4
Cost13056
\[\frac{\sqrt{0.5}}{t \cdot \pi} \]
Alternative 8
Error1.4
Cost13056
\[\frac{\frac{\sqrt{0.5}}{t}}{\pi} \]

Error

Reproduce

herbie shell --seed 2023010 
(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)))))