?

Average Error: 0.69% → 0.56%
Time: 11.0s
Precision: binary64
Cost: 14592

?

\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{1}{t \cdot \left(\pi \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right)} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v} \]
(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 (* t (* PI (sqrt (+ 2.0 (* 2.0 (* (* v v) -3.0)))))))
  (/ (- 1.0 (* (* v v) 5.0)) (- 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 (1.0 / (t * (((double) M_PI) * sqrt((2.0 + (2.0 * ((v * v) * -3.0))))))) * ((1.0 - ((v * v) * 5.0)) / (1.0 - (v * v)));
}

public static double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((Math.PI * t) * Math.sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
}
↓
public static double code(double v, double t) {
	return (1.0 / (t * (Math.PI * Math.sqrt((2.0 + (2.0 * ((v * v) * -3.0))))))) * ((1.0 - ((v * v) * 5.0)) / (1.0 - (v * v)));
}

def code(v, t):
	return (1.0 - (5.0 * (v * v))) / (((math.pi * t) * math.sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)))
↓
def code(v, t):
	return (1.0 / (t * (math.pi * math.sqrt((2.0 + (2.0 * ((v * v) * -3.0))))))) * ((1.0 - ((v * v) * 5.0)) / (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(1.0 / Float64(t * Float64(pi * sqrt(Float64(2.0 + Float64(2.0 * Float64(Float64(v * v) * -3.0))))))) * Float64(Float64(1.0 - Float64(Float64(v * v) * 5.0)) / Float64(1.0 - Float64(v * v))))
end

function tmp = code(v, t)
	tmp = (1.0 - (5.0 * (v * v))) / (((pi * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
end
↓
function tmp = code(v, t)
	tmp = (1.0 / (t * (pi * sqrt((2.0 + (2.0 * ((v * v) * -3.0))))))) * ((1.0 - ((v * v) * 5.0)) / (1.0 - (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[(1.0 / N[(t * N[(Pi * N[Sqrt[N[(2.0 + N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(v * v), $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{1}{t \cdot \left(\pi \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right)} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.69

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

  2. Applied egg-rr0.56

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

  3. Final simplification0.56

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

Alternatives

Alternative 1
Error0.69%
Cost14464
\[\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(1 - v \cdot v\right) \cdot \left(\left(t \cdot \pi\right) \cdot \sqrt{2 \cdot \left(1 + \left(v \cdot v\right) \cdot -3\right)}\right)} \]
Alternative 2
Error0.67%
Cost14464
\[\frac{\frac{1 + v \cdot \left(v \cdot -5\right)}{t \cdot \pi}}{\left(1 - v \cdot v\right) \cdot \sqrt{2 + 2 \cdot \left(v \cdot \left(v \cdot -3\right)\right)}} \]
Alternative 3
Error0.69%
Cost14336
\[\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \left(t \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}\right)\right)} \]
Alternative 4
Error0.69%
Cost14336
\[\frac{1 + v \cdot \left(v \cdot -5\right)}{\left(t \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}\right)} \]
Alternative 5
Error1.47%
Cost13184
\[\frac{1}{\left(t \cdot \pi\right) \cdot \sqrt{2}} \]
Alternative 6
Error1.42%
Cost13184
\[\frac{\frac{1}{\sqrt{2}}}{t \cdot \pi} \]
Alternative 7
Error0.96%
Cost13184
\[\frac{\frac{\frac{1}{\sqrt{2}}}{\pi}}{t} \]
Alternative 8
Error1.9%
Cost13056
\[\frac{\sqrt{0.5}}{t \cdot \pi} \]

Error

Reproduce?

herbie shell --seed 2023102 
(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)))))