Average Error: 1.0 → 0.0
Time: 5.1s
Precision: binary64
Cost: 20224
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
↓
\[\frac{1.3333333333333333 \cdot \frac{1}{\pi}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)} \]
(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))
↓
(FPCore (v)
 :precision binary64
 (/
  (* 1.3333333333333333 (/ 1.0 PI))
  (* (sqrt (fma v (* v -6.0) 2.0)) (- 1.0 (* v v)))))
double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
}
↓
double code(double v) {
	return (1.3333333333333333 * (1.0 / ((double) M_PI))) / (sqrt(fma(v, (v * -6.0), 2.0)) * (1.0 - (v * v)));
}

function code(v)
	return Float64(4.0 / Float64(Float64(Float64(3.0 * pi) * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v))))))
end
↓
function code(v)
	return Float64(Float64(1.3333333333333333 * Float64(1.0 / pi)) / Float64(sqrt(fma(v, Float64(v * -6.0), 2.0)) * Float64(1.0 - Float64(v * v))))
end

code[v_] := N[(4.0 / N[(N[(N[(3.0 * Pi), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[v_] := N[(N[(1.3333333333333333 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\frac{1.3333333333333333 \cdot \frac{1}{\pi}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)}

Error

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

  2. Applied egg-rr0.0

    \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(-6 \cdot v, v, 2\right)}}} \]

  3. Applied egg-rr0.0

    \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)}} \]

  4. Applied egg-rr0.0

    \[\leadsto \frac{\color{blue}{1.3333333333333333 \cdot \frac{1}{\pi}}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)} \]

  5. Final simplification0.0

    \[\leadsto \frac{1.3333333333333333 \cdot \frac{1}{\pi}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)} \]

Alternatives

Alternative 1
Error0.0
Cost20096
\[\frac{\frac{1.3333333333333333}{\pi}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(1 - v \cdot v\right)} \]
Alternative 2
Error0.6
Cost13440
\[\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)} \cdot \sqrt{0.5} \]
Alternative 3
Error0.6
Cost13056
\[\frac{1.3333333333333333}{\pi} \cdot \sqrt{0.5} \]

Error

Reproduce

herbie shell --seed 2022217 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))