Rosa's TurbineBenchmark

Average Error: 12.5 → 4.4

Time: 7.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\\ t_1 := 2 \cdot {r}^{-2}\\ \mathbf{if}\;r \leq -1.391648850866366 \cdot 10^{-38}:\\ \;\;\;\;t_1 - \mathsf{fma}\left(r, \left(r \cdot \left(w \cdot w\right)\right) \cdot t_0, 1.5\right)\\ \mathbf{elif}\;r \leq -4.72196167779232 \cdot 10^{-154}:\\ \;\;\;\;t_1 - 1.5\\ \mathbf{else}:\\ \;\;\;\;t_1 - \mathsf{fma}\left(r, {\left(w \cdot \sqrt{r \cdot t_0}\right)}^{2}, 1.5\right)\\ \end{array} \]

FPCore

(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
 (let* ((t_0 (/ (fma v -0.25 0.375) (- 1.0 v))) (t_1 (* 2.0 (pow r -2.0))))
   (if (<= r -1.391648850866366e-38)
     (- t_1 (fma r (* (* r (* w w)) t_0) 1.5))
     (if (<= r -4.72196167779232e-154)
       (- t_1 1.5)
       (- t_1 (fma r (pow (* w (sqrt (* r t_0))) 2.0) 1.5))))))

C

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) {
	double t_0 = fma(v, -0.25, 0.375) / (1.0 - v);
	double t_1 = 2.0 * pow(r, -2.0);
	double tmp;
	if (r <= -1.391648850866366e-38) {
		tmp = t_1 - fma(r, ((r * (w * w)) * t_0), 1.5);
	} else if (r <= -4.72196167779232e-154) {
		tmp = t_1 - 1.5;
	} else {
		tmp = t_1 - fma(r, pow((w * sqrt((r * t_0))), 2.0), 1.5);
	}
	return tmp;
}

Julia

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)
	t_0 = Float64(fma(v, -0.25, 0.375) / Float64(1.0 - v))
	t_1 = Float64(2.0 * (r ^ -2.0))
	tmp = 0.0
	if (r <= -1.391648850866366e-38)
		tmp = Float64(t_1 - fma(r, Float64(Float64(r * Float64(w * w)) * t_0), 1.5));
	elseif (r <= -4.72196167779232e-154)
		tmp = Float64(t_1 - 1.5);
	else
		tmp = Float64(t_1 - fma(r, (Float64(w * sqrt(Float64(r * t_0))) ^ 2.0), 1.5));
	end
	return tmp
end

Wolfram

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_] := Block[{t$95$0 = N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, -1.391648850866366e-38], N[(t$95$1 - N[(r * N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, -4.72196167779232e-154], N[(t$95$1 - 1.5), $MachinePrecision], N[(t$95$1 - N[(r * N[Power[N[(w * N[Sqrt[N[(r * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

1. Split input into 3 regimes

2. if r < -1.39164885086636609e-38

   1. Initial program 13.4

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

   2. Simplified 6.6

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

   3. Applied egg-rr 6.6

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

   4. Taylor expanded in r around 0 6.6

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

   5. Simplified 6.6

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

   if -1.39164885086636609e-38 < r < -4.72196167779231994e-154

   1. Initial program 12.4

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

   2. Simplified 11.7

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

   3. Applied egg-rr 11.5

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

   4. Taylor expanded in r around 0 6.4

      \[\leadsto 2 \cdot {r}^{-2} - \color{blue}{1.5} \]

   if -4.72196167779231994e-154 < r

   1. Initial program 11.9

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

   2. Simplified 7.8

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

   3. Applied egg-rr 7.7

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

   4. Taylor expanded in r around 0 7.7

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

   5. Simplified 7.7

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

   6. Applied egg-rr 2.3

      \[\leadsto 2 \cdot {r}^{-2} - \mathsf{fma}\left(r, \color{blue}{{\left(w \cdot \sqrt{r \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}}\right)}^{2}}, 1.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -1.391648850866366 \cdot 10^{-38}:\\ \;\;\;\;2 \cdot {r}^{-2} - \mathsf{fma}\left(r, \left(r \cdot \left(w \cdot w\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)\\ \mathbf{elif}\;r \leq -4.72196167779232 \cdot 10^{-154}:\\ \;\;\;\;2 \cdot {r}^{-2} - 1.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {r}^{-2} - \mathsf{fma}\left(r, {\left(w \cdot \sqrt{r \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}}\right)}^{2}, 1.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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))