Average Error: 0.0 → 0.0
Time: 2.1s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\sqrt{e^{2 \cdot \mathsf{log1p}\left(-v \cdot v\right)} \cdot \left(\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 0.125\right)} \]
(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
  (* (exp (* 2.0 (log1p (- (* v v))))) (* (+ 1.0 (* (* v v) -3.0)) 0.125))))
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((exp((2.0 * log1p(-(v * v)))) * ((1.0 + ((v * v) * -3.0)) * 0.125)));
}

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((Math.exp((2.0 * Math.log1p(-(v * v)))) * ((1.0 + ((v * v) * -3.0)) * 0.125)));
}

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((math.exp((2.0 * math.log1p(-(v * v)))) * ((1.0 + ((v * v) * -3.0)) * 0.125)))

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 sqrt(Float64(exp(Float64(2.0 * log1p(Float64(-Float64(v * v))))) * Float64(Float64(1.0 + Float64(Float64(v * v) * -3.0)) * 0.125)))
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[Sqrt[N[(N[Exp[N[(2.0 * N[Log[1 + (-N[(v * v), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision] * 0.125), $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)
↓
\sqrt{e^{2 \cdot \mathsf{log1p}\left(-v \cdot v\right)} \cdot \left(\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 0.125\right)}

Error

Bits error versus v

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. Applied egg-rr0.0

    \[\leadsto \color{blue}{\sqrt{{\left(1 - v \cdot v\right)}^{2} \cdot \left(\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 0.125\right)}} \]

  3. Applied egg-rr0.0

    \[\leadsto \sqrt{\color{blue}{e^{\left(2 \cdot \mathsf{log1p}\left(v \cdot \left(-v\right)\right)\right) \cdot 1}} \cdot \left(\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 0.125\right)} \]

  4. Final simplification0.0

    \[\leadsto \sqrt{e^{2 \cdot \mathsf{log1p}\left(-v \cdot v\right)} \cdot \left(\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 0.125\right)} \]

Reproduce

herbie shell --seed 2022162 
(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))))