Falkner and Boettcher, Appendix B, 2

Average Error: 0.0 → 0.0

Time: 1.9s

Precision: binary64

Math

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

↓

\[\frac{\left(1 - {v}^{4}\right) \cdot \sqrt{\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 0.125}}{1 + v \cdot v} \]

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
 (/
  (* (- 1.0 (pow v 4.0)) (sqrt (* (+ 1.0 (* (* v v) -3.0)) 0.125)))
  (+ 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 ((1.0 - pow(v, 4.0)) * sqrt(((1.0 + ((v * v) * -3.0)) * 0.125))) / (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 = ((1.0d0 - (v ** 4.0d0)) * sqrt(((1.0d0 + ((v * v) * (-3.0d0))) * 0.125d0))) / (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 ((1.0 - Math.pow(v, 4.0)) * Math.sqrt(((1.0 + ((v * v) * -3.0)) * 0.125))) / (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 ((1.0 - math.pow(v, 4.0)) * math.sqrt(((1.0 + ((v * v) * -3.0)) * 0.125))) / (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(Float64(1.0 - (v ^ 4.0)) * sqrt(Float64(Float64(1.0 + Float64(Float64(v * v) * -3.0)) * 0.125))) / 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 = ((1.0 - (v ^ 4.0)) * sqrt(((1.0 + ((v * v) * -3.0)) * 0.125))) / (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[(N[(1.0 - N[Power[v, 4.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision] * 0.125), $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

   \[\leadsto \color{blue}{\frac{\left(1 - {v}^{4}\right) \cdot \sqrt{\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 0.125}}{v \cdot v + 1}} \]

3. Final simplification 0.0

   \[\leadsto \frac{\left(1 - {v}^{4}\right) \cdot \sqrt{\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 0.125}}{1 + v \cdot v} \]

Reproduce

herbie shell --seed 2022192 
(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))))