Rosa's TurbineBenchmark

Average Error: 12.3 → 0.3

Time: 11.4s

Precision: binary64

Cost: 1728

Math

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

↓

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

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
 (+
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (* (* r w) (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r w))))
  -4.5))

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) {
	return ((3.0 + (2.0 / (r * r))) - ((r * w) * (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * w)))) + -4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

↓

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - ((r * w) * (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * w)))) + (-4.5d0)
end function

Java

public static 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;
}

↓

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((r * w) * (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * w)))) + -4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

↓

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - ((r * w) * (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * w)))) + -4.5

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)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(r * w) * Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * w)))) + -4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

↓

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - ((r * w) * (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * w)))) + -4.5;
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_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]

Derivation

1. Initial program 12.3

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

2. Simplified 8.2

   \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) + -4.5} \]
   Proof
   (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (+.f64 3 (*.f64 -2 v))) (/.f64 (-.f64 1 v) (*.f64 r (*.f64 r (*.f64 w w)))))) -9/2): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (+.f64 3 (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) v))) (/.f64 (-.f64 1 v) (*.f64 r (*.f64 r (*.f64 w w)))))) -9/2): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 3 (*.f64 2 v)))) (/.f64 (-.f64 1 v) (*.f64 r (*.f64 r (*.f64 w w)))))) -9/2): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (-.f64 1 v) (*.f64 r (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 w w) r)))))) -9/2): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (-.f64 1 v) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 w w) r) r))))) -9/2): 0 points increase in error, 0 points decrease in error
   (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)))) -9/2): 27 points increase in error, 5 points decrease in error
   (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))) (Rewrite<= metadata-eval (neg.f64 9/2))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))) 9/2)): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.3

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.8
Cost	1480

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{if}\;v \leq -6200000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 1:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.8
Cost	1352

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_2 := t_0 + \left(-1.5 - t_1 \cdot 0.25\right)\\ \mathbf{if}\;v \leq -6100000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 1:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	9.4
Cost	1088

\[\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \]

Alternative 4
Error	20.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;r \leq -1.15 \cdot 10^{+233}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]

Alternative 5
Error	21.1
Cost	448

\[\frac{2}{r \cdot r} + -1.5 \]

Alternative 6
Error	38.8
Cost	320

\[\frac{2}{r \cdot r} \]

Alternative 7
Error	38.8
Cost	320

\[\frac{\frac{2}{r}}{r} \]

Reproduce

herbie shell --seed 2022325 
(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))