Average Error: 0.0 → 0.0
Time: 12.2s
Precision: binary64
Cost: 26368
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\left(\left(-\mathsf{fma}\left(v, v, -1\right)\right) \cdot \sqrt{0.125}\right) \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right)} \]
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (* (* (- (fma v v -1.0)) (sqrt 0.125)) (sqrt (fma -3.0 (* v v) 1.0))))
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 (-fma(v, v, -1.0) * sqrt(0.125)) * sqrt(fma(-3.0, (v * v), 1.0));
}

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(-fma(v, v, -1.0)) * sqrt(0.125)) * sqrt(fma(-3.0, Float64(v * v), 1.0)))
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[(N[((-N[(v * v + -1.0), $MachinePrecision]) * N[Sqrt[0.125], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-3.0 * N[(v * v), $MachinePrecision] + 1.0), $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)
↓
\left(\left(-\mathsf{fma}\left(v, v, -1\right)\right) \cdot \sqrt{0.125}\right) \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right)}

Error

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. Simplified0.0

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

Alternatives

Alternative 1
Error0.0
Cost13888
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
Alternative 2
Error0.3
Cost7488
\[\left(\frac{\sqrt{2}}{4} \cdot \left(1 + -1.5 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]
Alternative 3
Error0.6
Cost6848
\[\sqrt{0.125} \cdot \left(1 - v \cdot v\right) \]

Error

Reproduce

herbie shell --seed 2023010 
(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))))