Average Error: 0.0 → 0.0
Time: 8.9s
Precision: binary64
Cost: 13888
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\frac{\sqrt{2} \cdot \left(1 - v \cdot v\right)}{\frac{4}{\sqrt{1 + \left(v \cdot v\right) \cdot -3}}} \]
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (/ (* (sqrt 2.0) (- 1.0 (* v v))) (/ 4.0 (sqrt (+ 1.0 (* (* v v) -3.0))))))
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 (sqrt(2.0) * (1.0 - (v * v))) / (4.0 / sqrt((1.0 + ((v * v) * -3.0))));
}

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 = (sqrt(2.0d0) * (1.0d0 - (v * v))) / (4.0d0 / sqrt((1.0d0 + ((v * v) * (-3.0d0)))))
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 (Math.sqrt(2.0) * (1.0 - (v * v))) / (4.0 / Math.sqrt((1.0 + ((v * v) * -3.0))));
}

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 (math.sqrt(2.0) * (1.0 - (v * v))) / (4.0 / math.sqrt((1.0 + ((v * v) * -3.0))))

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(sqrt(2.0) * Float64(1.0 - Float64(v * v))) / Float64(4.0 / sqrt(Float64(1.0 + Float64(Float64(v * v) * -3.0)))))
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 = (sqrt(2.0) * (1.0 - (v * v))) / (4.0 / sqrt((1.0 + ((v * v) * -3.0))));
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[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 / N[Sqrt[N[(1.0 + N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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{\sqrt{2} \cdot \left(1 - v \cdot v\right)}{\frac{4}{\sqrt{1 + \left(v \cdot v\right) \cdot -3}}}

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. Simplified0.0

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)} \]
    Proof
    (*.f64 (/.f64 (sqrt.f64 2) 4) (*.f64 (sqrt.f64 (-.f64 1 (*.f64 (*.f64 3 v) v))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sqrt.f64 2) 4) (*.f64 (sqrt.f64 (-.f64 1 (Rewrite<= associate-*r*_binary64 (*.f64 3 (*.f64 v v))))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (sqrt.f64 2) 4) (sqrt.f64 (-.f64 1 (*.f64 3 (*.f64 v v))))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost7360
\[\left(1 - v \cdot v\right) \cdot \left(0.25 \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}\right) \]
Alternative 2
Error0.3
Cost6976
\[\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.625\right) \]
Alternative 3
Error0.7
Cost6464
\[\sqrt{0.125} \]

Error

Reproduce

herbie shell --seed 2022343 
(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))))