Falkner and Boettcher, Appendix B, 2

Average Error: 0.0 → 0.0

Time: 4.5s

Precision: binary64

Cost: 7360

Math

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

↓

\[\left(0.25 \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}\right) \cdot \left(1 - v \cdot v\right) \]

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
 (* (* 0.25 (sqrt (+ 2.0 (* (* v v) -6.0)))) (- 1.0 (* v v))))

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 (0.25 * sqrt((2.0 + ((v * v) * -6.0)))) * (1.0 - (v * v));
}

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 = (0.25d0 * sqrt((2.0d0 + ((v * v) * (-6.0d0))))) * (1.0d0 - (v * v))
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 (0.25 * Math.sqrt((2.0 + ((v * v) * -6.0)))) * (1.0 - (v * v));
}

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 (0.25 * math.sqrt((2.0 + ((v * v) * -6.0)))) * (1.0 - (v * v))

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(Float64(0.25 * sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0)))) * Float64(1.0 - Float64(v * v)))
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 = (0.25 * sqrt((2.0 + ((v * v) * -6.0)))) * (1.0 - (v * v));
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[(0.25 * N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $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

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

3. Simplified 0.0

   \[\leadsto \color{blue}{\left(0.25 \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}\right)} \cdot \left(1 - v \cdot v\right) \]
   Proof

   [Start] 0.0	\[ \frac{1}{\frac{4}{\sqrt{2 + 2 \cdot \left(v \cdot \left(v \cdot -3\right)\right)}}} \cdot \left(1 - v \cdot v\right) \]
   associate-/r/ [=>] 0.0	\[ \color{blue}{\left(\frac{1}{4} \cdot \sqrt{2 + 2 \cdot \left(v \cdot \left(v \cdot -3\right)\right)}\right)} \cdot \left(1 - v \cdot v\right) \]
   metadata-eval [=>] 0.0	\[ \left(\color{blue}{0.25} \cdot \sqrt{2 + 2 \cdot \left(v \cdot \left(v \cdot -3\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]
   *-commutative [=>] 0.0	\[ \left(0.25 \cdot \sqrt{2 + \color{blue}{\left(v \cdot \left(v \cdot -3\right)\right) \cdot 2}}\right) \cdot \left(1 - v \cdot v\right) \]
   associate-*r* [=>] 0.0	\[ \left(0.25 \cdot \sqrt{2 + \color{blue}{\left(\left(v \cdot v\right) \cdot -3\right)} \cdot 2}\right) \cdot \left(1 - v \cdot v\right) \]
   associate-*l* [=>] 0.0	\[ \left(0.25 \cdot \sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-3 \cdot 2\right)}}\right) \cdot \left(1 - v \cdot v\right) \]
   metadata-eval [=>] 0.0	\[ \left(0.25 \cdot \sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}\right) \cdot \left(1 - v \cdot v\right) \]

4. Final simplification 0.0

   \[\leadsto \left(0.25 \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}\right) \cdot \left(1 - v \cdot v\right) \]

Alternatives

Alternative 1
Error	0.6
Cost	6976

\[\frac{\sqrt{0.125}}{\frac{1}{1 - v \cdot v}} \]

Alternative 2
Error	0.6
Cost	6848

\[\left(1 - v \cdot v\right) \cdot \sqrt{0.125} \]

Alternative 3
Error	0.6
Cost	6848

\[\frac{\sqrt{0.125}}{v \cdot v + 1} \]

Alternative 4
Error	0.6
Cost	6464

\[\sqrt{0.125} \]

Reproduce

herbie shell --seed 2022356 
(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))))