Average Error: 0.0 → 0.0
Time: 5.6s
Precision: binary64
Cost: 13888
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \left(1 - v \cdot v\right)\right) \]
(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) 4.0) (* (sqrt (+ 1.0 (* (* v v) -3.0))) (- 1.0 (* v v)))))
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) / 4.0) * (sqrt((1.0 + ((v * v) * -3.0))) * (1.0 - (v * v)));
}

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

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

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

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

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \left(1 - v \cdot v\right)\right)} \]
    Proof
    (*.f64 (/.f64 (sqrt.f64 2) 4) (*.f64 (sqrt.f64 (+.f64 1 (*.f64 (*.f64 v v) -3))) (-.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 (*.f64 (*.f64 v v) (Rewrite<= metadata-eval (neg.f64 3))))) (-.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<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 v v) 3))))) (-.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 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 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 (Rewrite<= sub-neg_binary64 (-.f64 1 (*.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)))): 1 points increase in error, 0 points decrease in error

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost13632
\[\left(1 - v \cdot v\right) \cdot \sqrt{0.125 \cdot \mathsf{fma}\left(v, v \cdot -3, 1\right)} \]
Alternative 2
Error0.3
Cost13376
\[\frac{\sqrt{2}}{\frac{4}{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}} \]
Alternative 3
Error0.3
Cost13248
\[\sqrt{0.125} \cdot \mathsf{fma}\left(v, v \cdot -2.5, 1\right) \]
Alternative 4
Error0.3
Cost6976
\[\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.625\right) \]
Alternative 5
Error0.6
Cost6848
\[\left(1 - v \cdot v\right) \cdot \sqrt{0.125} \]
Alternative 6
Error0.7
Cost6464
\[\sqrt{0.125} \]

Error

Reproduce

herbie shell --seed 2022337 
(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))))