Rosa's TurbineBenchmark

Average Error: 12.9 → 0.9

Time: 13.7s

Precision: binary64

Cost: 7816

Math

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

↓

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \left(\left(3 + 2 \cdot {r}^{-2}\right) + t_0 \cdot -0.25\right) + -4.5\\ \mathbf{if}\;v \leq -1.6909419061032551:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 2.0758199704787045 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + t_0 \cdot \left(-0.375 + \left(v + v \cdot v\right) \cdot -0.125\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* (* r w) (* r w)))
        (t_1 (+ (+ (+ 3.0 (* 2.0 (pow r -2.0))) (* t_0 -0.25)) -4.5)))
   (if (<= v -1.6909419061032551)
     t_1
     (if (<= v 2.0758199704787045e-18)
       (+
        (+ (+ 3.0 (/ 2.0 (* r r))) (* t_0 (+ -0.375 (* (+ v (* v v)) -0.125))))
        -4.5)
       t_1))))

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 = (r * w) * (r * w);
	double t_1 = ((3.0 + (2.0 * pow(r, -2.0))) + (t_0 * -0.25)) + -4.5;
	double tmp;
	if (v <= -1.6909419061032551) {
		tmp = t_1;
	} else if (v <= 2.0758199704787045e-18) {
		tmp = ((3.0 + (2.0 / (r * r))) + (t_0 * (-0.375 + ((v + (v * v)) * -0.125)))) + -4.5;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = ((3.0d0 + (2.0d0 * (r ** (-2.0d0)))) + (t_0 * (-0.25d0))) + (-4.5d0)
    if (v <= (-1.6909419061032551d0)) then
        tmp = t_1
    else if (v <= 2.0758199704787045d-18) then
        tmp = ((3.0d0 + (2.0d0 / (r * r))) + (t_0 * ((-0.375d0) + ((v + (v * v)) * (-0.125d0))))) + (-4.5d0)
    else
        tmp = t_1
    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 = (r * w) * (r * w);
	double t_1 = ((3.0 + (2.0 * Math.pow(r, -2.0))) + (t_0 * -0.25)) + -4.5;
	double tmp;
	if (v <= -1.6909419061032551) {
		tmp = t_1;
	} else if (v <= 2.0758199704787045e-18) {
		tmp = ((3.0 + (2.0 / (r * r))) + (t_0 * (-0.375 + ((v + (v * v)) * -0.125)))) + -4.5;
	} else {
		tmp = t_1;
	}
	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 = (r * w) * (r * w)
	t_1 = ((3.0 + (2.0 * math.pow(r, -2.0))) + (t_0 * -0.25)) + -4.5
	tmp = 0
	if v <= -1.6909419061032551:
		tmp = t_1
	elif v <= 2.0758199704787045e-18:
		tmp = ((3.0 + (2.0 / (r * r))) + (t_0 * (-0.375 + ((v + (v * v)) * -0.125)))) + -4.5
	else:
		tmp = t_1
	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(r * w) * Float64(r * w))
	t_1 = Float64(Float64(Float64(3.0 + Float64(2.0 * (r ^ -2.0))) + Float64(t_0 * -0.25)) + -4.5)
	tmp = 0.0
	if (v <= -1.6909419061032551)
		tmp = t_1;
	elseif (v <= 2.0758199704787045e-18)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(t_0 * Float64(-0.375 + Float64(Float64(v + Float64(v * v)) * -0.125)))) + -4.5);
	else
		tmp = t_1;
	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 = (r * w) * (r * w);
	t_1 = ((3.0 + (2.0 * (r ^ -2.0))) + (t_0 * -0.25)) + -4.5;
	tmp = 0.0;
	if (v <= -1.6909419061032551)
		tmp = t_1;
	elseif (v <= 2.0758199704787045e-18)
		tmp = ((3.0 + (2.0 / (r * r))) + (t_0 * (-0.375 + ((v + (v * v)) * -0.125)))) + -4.5;
	else
		tmp = t_1;
	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[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * -0.25), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]}, If[LessEqual[v, -1.6909419061032551], t$95$1, If[LessEqual[v, 2.0758199704787045e-18], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(-0.375 + N[(N[(v + N[(v * v), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.690941906103255 or 2.0758199704787045e-18 < v

   1. Initial program 16.9

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

   2. Applied egg-rr 0.5

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

   3. Taylor expanded in v around inf 18.2

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

   4. Simplified 1.4

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

   5. Applied egg-rr 1.3

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

   if -1.690941906103255 < v < 2.0758199704787045e-18

   1. Initial program 8.5

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

   2. Applied egg-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. Applied egg-rr 0.4

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

   4. Taylor expanded in v around 0 17.7

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

   5. Simplified 0.4

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \left(0.125 \cdot \left(v + v \cdot v\right) + 0.375\right)}\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.6909419061032551:\\ \;\;\;\;\left(\left(3 + 2 \cdot {r}^{-2}\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -4.5\\ \mathbf{elif}\;v \leq 2.0758199704787045 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(-0.375 + \left(v + v \cdot v\right) \cdot -0.125\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + 2 \cdot {r}^{-2}\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -4.5\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
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	0.4
Cost	14336

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

Alternative 3
Error	0.4
Cost	8128

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

Alternative 4
Error	0.4
Cost	8128

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

Alternative 5
Error	1.2
Cost	1992

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t_1 + t_0 \cdot -0.25\right) + -4.5\\ \mathbf{if}\;v \leq -17260680353481773000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 2.0758199704787045 \cdot 10^{-18}:\\ \;\;\;\;\left(t_1 + t_0 \cdot \left(-0.375 + \left(v + v \cdot v\right) \cdot -0.125\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	7.7
Cost	1612

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 3 + t_0\\ t_2 := \left(t_1 + r \cdot \left(\left(w \cdot w\right) \cdot \left(r \cdot -0.25\right)\right)\right) + -4.5\\ \mathbf{if}\;v \leq -17260680353481773000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 1.9805378681870293 \cdot 10^{+80}:\\ \;\;\;\;\left(t_1 + w \cdot \left(r \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\right) + -4.5\\ \mathbf{elif}\;v \leq 3.4696281912675885 \cdot 10^{+301}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	9.3
Cost	1480

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot -0.375\\ t_1 := \left(r \cdot w\right) \cdot t_0 + -1.5\\ \mathbf{if}\;r \leq -1 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 10^{+160}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + w \cdot \left(r \cdot t_0\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	3.1
Cost	1480

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -4.5\\ \mathbf{if}\;v \leq -17260680353481773000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 2.0758199704787045 \cdot 10^{-18}:\\ \;\;\;\;\left(t_0 + w \cdot \left(r \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	1.2
Cost	1480

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t_1 + t_0 \cdot -0.25\right) + -4.5\\ \mathbf{if}\;v \leq -17260680353481773000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 2.0758199704787045 \cdot 10^{-18}:\\ \;\;\;\;\left(t_1 + t_0 \cdot -0.375\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	12.4
Cost	968

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right) + -1.5\\ \mathbf{if}\;r \leq -6 \cdot 10^{+19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 59:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	21.0
Cost	448

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

Alternative 12
Error	21.0
Cost	448

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

Alternative 13
Error	38.4
Cost	320

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

Reproduce

herbie shell --seed 2022217 
(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))