?

Average Error: 1.0 → 0.0
Time: 14.4s
Precision: binary64
Cost: 13824

?

\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
↓
\[\frac{\frac{1.3333333333333333}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}}}{\pi \cdot \left(1 - v \cdot v\right)} \]
(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 (sqrt (- 2.0 (* v (* v 6.0)))))
  (* PI (- 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 / sqrt((2.0 - (v * (v * 6.0))))) / (((double) M_PI) * (1.0 - (v * v)));
}

public static double code(double v) {
	return 4.0 / (((3.0 * Math.PI) * (1.0 - (v * v))) * Math.sqrt((2.0 - (6.0 * (v * v)))));
}
↓
public static double code(double v) {
	return (1.3333333333333333 / Math.sqrt((2.0 - (v * (v * 6.0))))) / (Math.PI * (1.0 - (v * v)));
}

def code(v):
	return 4.0 / (((3.0 * math.pi) * (1.0 - (v * v))) * math.sqrt((2.0 - (6.0 * (v * v)))))
↓
def code(v):
	return (1.3333333333333333 / math.sqrt((2.0 - (v * (v * 6.0))))) / (math.pi * (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 / sqrt(Float64(2.0 - Float64(v * Float64(v * 6.0))))) / Float64(pi * Float64(1.0 - Float64(v * v))))
end

function tmp = code(v)
	tmp = 4.0 / (((3.0 * pi) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
end
↓
function tmp = code(v)
	tmp = (1.3333333333333333 / sqrt((2.0 - (v * (v * 6.0))))) / (pi * (1.0 - (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[Sqrt[N[(2.0 - N[(v * N[(v * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * 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{\frac{1.3333333333333333}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}}}{\pi \cdot \left(1 - v \cdot v\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

    \[\leadsto \color{blue}{\frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} + 0} \]

  3. Simplified0.0

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

    [Start]0.0

    \[ \frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} + 0 \]

    rational.json-simplify-4 [=>]0.0

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

    rational.json-simplify-47 [=>]0.0

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

    rational.json-simplify-44 [=>]0.0

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

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.5
Cost13696
\[\frac{4}{\pi} \cdot \frac{1}{3 \cdot \sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]
Alternative 2
Error0.5
Cost13568
\[\frac{4}{\frac{\pi}{\frac{0.3333333333333333}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}}}} \]
Alternative 3
Error0.5
Cost13568
\[\frac{\frac{4}{3 \cdot \pi}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]
Alternative 4
Error0.5
Cost13440
\[\frac{1.3333333333333333}{\pi \cdot \sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]
Alternative 5
Error1.5
Cost13056
\[1.3333333333333333 \cdot \frac{\sqrt{0.5}}{\pi} \]
Alternative 6
Error0.5
Cost13056
\[\frac{\frac{1.3333333333333333}{\pi}}{\sqrt{2}} \]

Error

Reproduce?

herbie shell --seed 2023068 
(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)))))))