Falkner and Boettcher, Appendix B, 2

Average Error: 0.0 → 0.0

Time: 1.4min

Precision: binary64

Cost: 26368

Math

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

↓

\[\sqrt{\mathsf{fma}\left(-3 \cdot v, v, 1\right)} \cdot \left(\sqrt{0.125} \cdot \left(-\mathsf{fma}\left(v, v, -1\right)\right)\right) \]

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
 (* (sqrt (fma (* -3.0 v) v 1.0)) (* (sqrt 0.125) (- (fma v v -1.0)))))

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(fma((-3.0 * v), v, 1.0)) * (sqrt(0.125) * -fma(v, v, -1.0));
}

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(fma(Float64(-3.0 * v), v, 1.0)) * Float64(sqrt(0.125) * Float64(-fma(v, v, -1.0))))
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[(N[(-3.0 * v), $MachinePrecision] * v + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.125], $MachinePrecision] * (-N[(v * v + -1.0), $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. Simplified 0.0

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-3 \cdot v, v, 1\right)} \cdot \left(\sqrt{0.125} \cdot \left(-\mathsf{fma}\left(v, v, -1\right)\right)\right)} \]
   Proof

Alternatives

Alternative 1
Error	0.0
Cost	13888

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

Alternative 2
Error	0.3
Cost	7488

\[\left(\frac{\sqrt{2}}{4} \cdot \left(1 + -1.5 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

Alternative 3
Error	0.7
Cost	6848

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

Alternative 4
Error	53.6
Cost	6784

\[e \cdot \left(1 - v \cdot v\right) \]

Reproduce

herbie shell --seed 2023033 
(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))))