Falkner and Boettcher, Appendix B, 2

Average Error: 0.0 → 0.0

Time: 4.1s

Precision: binary64

Cost: 7360

Math

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

↓

\[\sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot \frac{1 - v \cdot v}{4} \]

FPCore

(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 (* -6.0 (* v v)))) (/ (- 1.0 (* v v)) 4.0)))

C

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 + (-6.0 * (v * v)))) * ((1.0 - (v * v)) / 4.0);
}

Fortran

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 + ((-6.0d0) * (v * v)))) * ((1.0d0 - (v * v)) / 4.0d0)
end function

Java

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 + (-6.0 * (v * v)))) * ((1.0 - (v * v)) / 4.0);
}

Python

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 + (-6.0 * (v * v)))) * ((1.0 - (v * v)) / 4.0)

Julia

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(sqrt(Float64(2.0 + Float64(-6.0 * Float64(v * v)))) * Float64(Float64(1.0 - Float64(v * v)) / 4.0))
end

MATLAB

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 + (-6.0 * (v * v)))) * ((1.0 - (v * v)) / 4.0);
end

Wolfram

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[Sqrt[N[(2.0 + N[(-6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]

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-rr 0.0

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

3. Simplified 0.0

   \[\leadsto \color{blue}{\sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot \frac{1 - v \cdot v}{4}} \]
   Proof

   [Start] 0.0	\[ \frac{1 - v \cdot v}{\frac{4}{\sqrt{2 \cdot \left(1 + \left(-3 \cdot v\right) \cdot v\right)}}} \]
   associate-/r/ [=>] 0.0	\[ \color{blue}{\frac{1 - v \cdot v}{4} \cdot \sqrt{2 \cdot \left(1 + \left(-3 \cdot v\right) \cdot v\right)}} \]
   *-commutative [=>] 0.0	\[ \color{blue}{\sqrt{2 \cdot \left(1 + \left(-3 \cdot v\right) \cdot v\right)} \cdot \frac{1 - v \cdot v}{4}} \]
   distribute-lft-in [=>] 0.0	\[ \sqrt{\color{blue}{2 \cdot 1 + 2 \cdot \left(\left(-3 \cdot v\right) \cdot v\right)}} \cdot \frac{1 - v \cdot v}{4} \]
   metadata-eval [=>] 0.0	\[ \sqrt{\color{blue}{2} + 2 \cdot \left(\left(-3 \cdot v\right) \cdot v\right)} \cdot \frac{1 - v \cdot v}{4} \]
   associate-*l* [=>] 0.0	\[ \sqrt{2 + 2 \cdot \color{blue}{\left(-3 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{1 - v \cdot v}{4} \]
   associate-*r* [=>] 0.0	\[ \sqrt{2 + \color{blue}{\left(2 \cdot -3\right) \cdot \left(v \cdot v\right)}} \cdot \frac{1 - v \cdot v}{4} \]
   metadata-eval [=>] 0.0	\[ \sqrt{2 + \color{blue}{-6} \cdot \left(v \cdot v\right)} \cdot \frac{1 - v \cdot v}{4} \]

4. Final simplification 0.0

   \[\leadsto \sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot \frac{1 - v \cdot v}{4} \]

Alternatives

Alternative 1
Error	0.3
Cost	6976

\[\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.625\right) \]

Alternative 2
Error	0.7
Cost	6592

\[\sqrt{2} \cdot 0.25 \]

Reproduce

herbie shell --seed 2023060 
(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))))