Average Error: 0.0 → 0.0
Time: 3.2s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\left|\left(\frac{\sqrt{2}}{4} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}\right| \]
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (fabs
  (* (* (/ (sqrt 2.0) 4.0) (- 1.0 (* v v))) (sqrt (fma (* v v) -3.0 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 fabs((((sqrt(2.0) / 4.0) * (1.0 - (v * v))) * sqrt(fma((v * v), -3.0, 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 abs(Float64(Float64(Float64(sqrt(2.0) / 4.0) * Float64(1.0 - Float64(v * v))) * sqrt(fma(Float64(v * v), -3.0, 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[Abs[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision]], $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(\frac{\sqrt{2}}{4} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}\right|

Error

Bits error versus v

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

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

  3. Applied egg-rr0.0

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

  4. Applied egg-rr0.0

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

  5. Final simplification0.0

    \[\leadsto \left|\left(\frac{\sqrt{2}}{4} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}\right| \]

Reproduce

herbie shell --seed 2022131 
(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))))