?

Average Error: 0.0 → 0.0
Time: 7.0s
Precision: binary64
Cost: 7616

?

\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\frac{\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \left(-2 - \left(3 \cdot \left(v \cdot v\right) + -3\right)\right)}}{4} \]
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (/ (* (- 1.0 (* v v)) (sqrt (* 2.0 (- -2.0 (+ (* 3.0 (* v v)) -3.0))))) 4.0))
double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}
↓
double code(double v) {
	return ((1.0 - (v * v)) * sqrt((2.0 * (-2.0 - ((3.0 * (v * v)) + -3.0))))) / 4.0;
}

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function
↓
real(8) function code(v)
    real(8), intent (in) :: v
    code = ((1.0d0 - (v * v)) * sqrt((2.0d0 * ((-2.0d0) - ((3.0d0 * (v * v)) + (-3.0d0)))))) / 4.0d0
end function

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

def code(v):
	return ((math.sqrt(2.0) / 4.0) * math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v))
↓
def code(v):
	return ((1.0 - (v * v)) * math.sqrt((2.0 * (-2.0 - ((3.0 * (v * v)) + -3.0))))) / 4.0

function code(v)
	return Float64(Float64(Float64(sqrt(2.0) / 4.0) * sqrt(Float64(1.0 - Float64(3.0 * Float64(v * v))))) * Float64(1.0 - Float64(v * v)))
end
↓
function code(v)
	return Float64(Float64(Float64(1.0 - Float64(v * v)) * sqrt(Float64(2.0 * Float64(-2.0 - Float64(Float64(3.0 * Float64(v * v)) + -3.0))))) / 4.0)
end

function tmp = code(v)
	tmp = ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
end
↓
function tmp = code(v)
	tmp = ((1.0 - (v * v)) * sqrt((2.0 * (-2.0 - ((3.0 * (v * v)) + -3.0))))) / 4.0;
end

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

\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \left(-2 - \left(3 \cdot \left(v \cdot v\right) + -3\right)\right)}}{4}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.0

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

  2. Applied egg-rr0.0

    \[\leadsto \color{blue}{\frac{\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}{4}} \]

  3. Applied egg-rr0.0

    \[\leadsto \frac{\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \color{blue}{\left(\left(1 - \left(3 \cdot \left(v \cdot v\right) + -1\right)\right) + -1\right)}}}{4} \]

  4. Applied egg-rr0.0

    \[\leadsto \frac{\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\left(1 - \left(3 \cdot \left(v \cdot v\right) + -2\right)\right) + -1\right)} + -1\right)}}{4} \]

  5. Applied egg-rr0.0

    \[\leadsto \frac{\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \color{blue}{\left(-2 - \left(3 \cdot \left(v \cdot v\right) + -3\right)\right)}}}{4} \]

  6. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost7424
\[\frac{\left(1 - v \cdot v\right) \cdot \sqrt{-\left(-2 + v \cdot \left(v \cdot 6\right)\right)}}{4} \]
Alternative 2
Error0.7
Cost6976
\[\frac{\sqrt{2} \cdot \left(1 - v \cdot v\right)}{4} \]
Alternative 3
Error0.7
Cost6592
\[0.25 \cdot \sqrt{2} \]

Error

Reproduce?

herbie shell --seed 2023067 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))