Average Error: 12.7 → 0.3
Time: 8.4s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
↓
\[\frac{2}{r \cdot r} - \left(\frac{{\left(r \cdot w\right)}^{2}}{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}} + 1.5\right) \]
(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
 (-
  (/ 2.0 (* r r))
  (+ (/ (pow (* r w) 2.0) (/ (- 1.0 v) (fma v -0.25 0.375))) 1.5)))
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 (2.0 / (r * r)) - ((pow((r * w), 2.0) / ((1.0 - v) / fma(v, -0.25, 0.375))) + 1.5);
}

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(2.0 / Float64(r * r)) - Float64(Float64((Float64(r * w) ^ 2.0) / Float64(Float64(1.0 - v) / fma(v, -0.25, 0.375))) + 1.5))
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[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $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
↓
\frac{2}{r \cdot r} - \left(\frac{{\left(r \cdot w\right)}^{2}}{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}} + 1.5\right)

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.7

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

  2. Simplified8.5

    \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(r, \left(r \cdot \left(w \cdot w\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)} \]

  3. Applied egg-rr8.5

    \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(r, \color{blue}{\frac{r \cdot \left(w \cdot w\right)}{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}}, 1.5\right) \]

  4. Applied egg-rr2.1

    \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(r, \color{blue}{\frac{r \cdot w}{\sqrt{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}} \cdot \frac{w}{\sqrt{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}}}, 1.5\right) \]

  5. Applied egg-rr0.3

    \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{{\left(r \cdot w\right)}^{2}}{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}} + 1.5\right)} \]

  6. Final simplification0.3

    \[\leadsto \frac{2}{r \cdot r} - \left(\frac{{\left(r \cdot w\right)}^{2}}{\frac{1 - v}{\mathsf{fma}\left(v, -0.25, 0.375\right)}} + 1.5\right) \]

Reproduce

herbie shell --seed 2022133 
(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))