Average Error: 0.4 → 0.1
Time: 4.4s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \cdot \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi}}{t}}{1 - v \cdot v} \]
(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
 (/
  (/ (* (/ 1.0 (sqrt (fma (* v v) -6.0 2.0))) (/ (fma v (* v -5.0) 1.0) PI)) t)
  (- 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 (((1.0 / sqrt(fma((v * v), -6.0, 2.0))) * (fma(v, (v * -5.0), 1.0) / ((double) M_PI))) / t) / (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(1.0 / sqrt(fma(Float64(v * v), -6.0, 2.0))) * Float64(fma(v, Float64(v * -5.0), 1.0) / pi)) / t) / 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[(1.0 / N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] / t), $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{1}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \cdot \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi}}{t}}{1 - v \cdot v}

Error

Bits error versus v

Bits error versus t

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. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t \cdot \left(\pi \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}\right)}}{1 - v \cdot v}} \]

  3. Applied egg-rr0.3

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t} \cdot \frac{1}{\pi \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}}{1 - v \cdot v} \]

  4. Applied egg-rr0.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{t}}}{1 - v \cdot v} \]

  5. Applied egg-rr0.1

    \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \cdot \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi}}}{t}}{1 - v \cdot v} \]

  6. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \cdot \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi}}{t}}{1 - v \cdot v} \]

Reproduce

herbie shell --seed 2022151 
(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)))))