Rosa's TurbineBenchmark

Average Error: 13.1 → 0.3

Time: 15.3s

Precision: binary64

Cost: 8328

Math

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

↓

\[\begin{array}{l} t_0 := \left(\left(3 + 2 \cdot {r}^{-2}\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -4.5\\ \mathbf{if}\;v \leq -8.533760124110259 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 247618436451.14963:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{{\left(r \cdot w\right)}^{2} \cdot \left(0.125 \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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
         (+
          (+ (+ 3.0 (* 2.0 (pow r -2.0))) (* (* (* r w) (* r w)) -0.25))
          -4.5)))
   (if (<= v -8.533760124110259e+22)
     t_0
     (if (<= v 247618436451.14963)
       (+
        (-
         (+ 3.0 (/ 2.0 (* r r)))
         (/ (* (pow (* r w) 2.0) (* 0.125 (+ 3.0 (* v -2.0)))) (- 1.0 v)))
        -4.5)
       t_0))))

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 = ((3.0 + (2.0 * pow(r, -2.0))) + (((r * w) * (r * w)) * -0.25)) + -4.5;
	double tmp;
	if (v <= -8.533760124110259e+22) {
		tmp = t_0;
	} else if (v <= 247618436451.14963) {
		tmp = ((3.0 + (2.0 / (r * r))) - ((pow((r * w), 2.0) * (0.125 * (3.0 + (v * -2.0)))) / (1.0 - v))) + -4.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((3.0d0 + (2.0d0 * (r ** (-2.0d0)))) + (((r * w) * (r * w)) * (-0.25d0))) + (-4.5d0)
    if (v <= (-8.533760124110259d+22)) then
        tmp = t_0
    else if (v <= 247618436451.14963d0) then
        tmp = ((3.0d0 + (2.0d0 / (r * r))) - ((((r * w) ** 2.0d0) * (0.125d0 * (3.0d0 + (v * (-2.0d0))))) / (1.0d0 - v))) + (-4.5d0)
    else
        tmp = t_0
    end if
    code = tmp
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) {
	double t_0 = ((3.0 + (2.0 * Math.pow(r, -2.0))) + (((r * w) * (r * w)) * -0.25)) + -4.5;
	double tmp;
	if (v <= -8.533760124110259e+22) {
		tmp = t_0;
	} else if (v <= 247618436451.14963) {
		tmp = ((3.0 + (2.0 / (r * r))) - ((Math.pow((r * w), 2.0) * (0.125 * (3.0 + (v * -2.0)))) / (1.0 - v))) + -4.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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):
	t_0 = ((3.0 + (2.0 * math.pow(r, -2.0))) + (((r * w) * (r * w)) * -0.25)) + -4.5
	tmp = 0
	if v <= -8.533760124110259e+22:
		tmp = t_0
	elif v <= 247618436451.14963:
		tmp = ((3.0 + (2.0 / (r * r))) - ((math.pow((r * w), 2.0) * (0.125 * (3.0 + (v * -2.0)))) / (1.0 - v))) + -4.5
	else:
		tmp = t_0
	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(Float64(Float64(3.0 + Float64(2.0 * (r ^ -2.0))) + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25)) + -4.5)
	tmp = 0.0
	if (v <= -8.533760124110259e+22)
		tmp = t_0;
	elseif (v <= 247618436451.14963)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64((Float64(r * w) ^ 2.0) * Float64(0.125 * Float64(3.0 + Float64(v * -2.0)))) / Float64(1.0 - v))) + -4.5);
	else
		tmp = t_0;
	end
	return tmp
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_2 = code(v, w, r)
	t_0 = ((3.0 + (2.0 * (r ^ -2.0))) + (((r * w) * (r * w)) * -0.25)) + -4.5;
	tmp = 0.0;
	if (v <= -8.533760124110259e+22)
		tmp = t_0;
	elseif (v <= 247618436451.14963)
		tmp = ((3.0 + (2.0 / (r * r))) - ((((r * w) ^ 2.0) * (0.125 * (3.0 + (v * -2.0)))) / (1.0 - v))) + -4.5;
	else
		tmp = t_0;
	end
	tmp_2 = 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[(N[(3.0 + N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]}, If[LessEqual[v, -8.533760124110259e+22], t$95$0, If[LessEqual[v, 247618436451.14963], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if v < -8.533760124110259e22 or 247618436451.149628 < v

   1. Initial program 17.7

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

   2. Applied egg-rr 0.3

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

   3. Taylor expanded in v around inf 17.6

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

   4. Simplified 0.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot 0.25}\right) - 4.5 \]
      Proof
      (*.f64 (*.f64 (*.f64 w r) (*.f64 w r)) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 w w) (*.f64 r r))) 1/4): 97 points increase in error, 46 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) (*.f64 r r)) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 w 2) (Rewrite<= unpow2_binary64 (pow.f64 r 2))) 1/4): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/4 (*.f64 (pow.f64 w 2) (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.2

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

   if -8.533760124110259e22 < v < 247618436451.149628

   1. Initial program 8.9

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

   2. Applied egg-rr 0.4

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{{\left(w \cdot r\right)}^{2}}}{1 - v}\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -8.533760124110259 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(3 + 2 \cdot {r}^{-2}\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -4.5\\ \mathbf{elif}\;v \leq 247618436451.14963:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{{\left(r \cdot w\right)}^{2} \cdot \left(0.125 \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + 2 \cdot {r}^{-2}\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -4.5\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	14336

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

Alternative 2
Error	0.3
Cost	7816

\[\begin{array}{l} t_0 := \left(\left(3 + 2 \cdot {r}^{-2}\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -4.5\\ \mathbf{if}\;v \leq -8.533760124110259 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 247618436451.14963:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.375 - v \cdot -0.25\right)\right)}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1992

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -4.5\\ \mathbf{if}\;v \leq -8.533760124110259 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 247618436451.14963:\\ \;\;\;\;\left(t_0 + \frac{\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.375 - v \cdot -0.25\right)\right)}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	10.5
Cost	1736

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 + \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right) \cdot -0.375\right) + -4.5\\ \mathbf{if}\;w \cdot w \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \cdot w \leq 4 \cdot 10^{+289}:\\ \;\;\;\;\left(t_0 - \frac{r \cdot \left(w \cdot w\right)}{\frac{2.6666666666666665}{r}}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.6
Cost	1608

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := \left(\left(3 + t_1\right) + t_0 \cdot -0.25\right) + -4.5\\ \mathbf{if}\;v \leq -27312.246903103573:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 0.028308313978675108:\\ \;\;\;\;t_0 \cdot \left(-0.375 - 0.125 \cdot v\right) + \left(t_1 + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	2.8
Cost	1480

\[\begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t_1 + r \cdot \left(-0.25 \cdot t_0\right)\right) + -4.5\\ \mathbf{if}\;v \leq -27312.246903103573:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 0.028308313978675108:\\ \;\;\;\;\left(t_1 + t_0 \cdot \left(r \cdot -0.375\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	1.8
Cost	1480

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -4.5\\ \mathbf{if}\;v \leq -5.3638706833169945 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 0.028308313978675108:\\ \;\;\;\;\left(t_0 + \left(w \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot -0.375\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	0.7
Cost	1480

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t_1 + t_0 \cdot -0.25\right) + -4.5\\ \mathbf{if}\;v \leq -27312.246903103573:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 0.028308313978675108:\\ \;\;\;\;\left(t_1 + t_0 \cdot -0.375\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	9.2
Cost	1476

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{+289}:\\ \;\;\;\;\left(t_0 + \left(w \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot -0.375\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right) \cdot -0.375\right) + -4.5\\ \end{array} \]

Alternative 10
Error	9.2
Cost	1476

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{-61}:\\ \;\;\;\;\left(t_0 + \left(w \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot -0.375\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right)\right)\right) + -4.5\\ \end{array} \]

Alternative 11
Error	11.8
Cost	1216

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

Alternative 12
Error	21.1
Cost	448

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

Alternative 13
Error	38.1
Cost	320

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

Reproduce

herbie shell --seed 2022312 
(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))