Average Error: 12.5 → 0.2
Time: 7.2s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
↓
\[\left(3 + \mathsf{fma}\left({r}^{-2}, 2, {\left(r \cdot w\right)}^{2} \cdot \frac{-\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right)\right) + -4.5 \]
(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))
↓
(FPCore (v w r)
 :precision binary64
 (+
  (+
   3.0
   (fma
    (pow r -2.0)
    2.0
    (* (pow (* r w) 2.0) (/ (- (fma v -0.25 0.375)) (- 1.0 v)))))
  -4.5))
double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}
↓
double code(double v, double w, double r) {
	return (3.0 + fma(pow(r, -2.0), 2.0, (pow((r * w), 2.0) * (-fma(v, -0.25, 0.375) / (1.0 - v))))) + -4.5;
}

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end
↓
function code(v, w, r)
	return Float64(Float64(3.0 + fma((r ^ -2.0), 2.0, Float64((Float64(r * w) ^ 2.0) * Float64(Float64(-fma(v, -0.25, 0.375)) / Float64(1.0 - v))))) + -4.5)
end

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]
↓
code[v_, w_, r_] := N[(N[(3.0 + N[(N[Power[r, -2.0], $MachinePrecision] * 2.0 + N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] * N[((-N[(v * -0.25 + 0.375), $MachinePrecision]) / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]

\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5
↓
\left(3 + \mathsf{fma}\left({r}^{-2}, 2, {\left(r \cdot w\right)}^{2} \cdot \frac{-\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right)\right) + -4.5

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.5

    \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

  2. Applied egg-rr0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.375 + \left(v \cdot -2\right) \cdot 0.125}{1 - v} \cdot {\left(w \cdot r\right)}^{2}}\right) - 4.5 \]

  3. Applied egg-rr0.2

    \[\leadsto \color{blue}{\left(3 + \left(2 \cdot {r}^{-2} - {\left(r \cdot w\right)}^{2} \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right)\right)} - 4.5 \]

  4. Applied egg-rr0.3

    \[\leadsto \left(3 + \left(2 \cdot {r}^{-2} - {\left(r \cdot w\right)}^{2} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right)\right)}\right)\right) - 4.5 \]

  5. Applied egg-rr0.2

    \[\leadsto \left(3 + \color{blue}{\mathsf{fma}\left({r}^{-2}, 2, \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(-{\left(r \cdot w\right)}^{2}\right)\right)}\right) - 4.5 \]

  6. Final simplification0.2

    \[\leadsto \left(3 + \mathsf{fma}\left({r}^{-2}, 2, {\left(r \cdot w\right)}^{2} \cdot \frac{-\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right)\right) + -4.5 \]

Reproduce

herbie shell --seed 2022170 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))