Average Error: 0.4 → 0.1
Time: 5.7s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi \cdot \left(\sqrt{2} \cdot \left(1 - v \cdot v\right)\right)}}{t} \cdot \sqrt{\frac{1}{1 - \left(v \cdot v\right) \cdot 3}} \]
(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 (* (sqrt 2.0) (- 1.0 (* v v))))) t)
  (sqrt (/ 1.0 (- 1.0 (* (* v v) 3.0))))))
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) * (sqrt(2.0) * (1.0 - (v * v))))) / t) * sqrt((1.0 / (1.0 - ((v * v) * 3.0))));
}

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

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. Taylor expanded in t around 0 0.4

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

  3. Simplified0.4

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

  4. Applied *-un-lft-identity_binary640.4

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

  5. Applied times-frac_binary640.3

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

  6. Simplified0.3

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

  7. Applied associate-*l/_binary640.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2022130 
(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)))))