Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.7%

Time: 11.7s

Alternatives: 5

Speedup: 1.7×

• 53.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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))

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;
}

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

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;
}

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

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

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

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]

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

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))

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;
}

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

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;
}

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

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

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

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]

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (+
   3.0
   (-
    (/ 2.0 (* r r))
    (/
     (* 0.125 (+ 3.0 (* -2.0 v)))
     (* (/ (/ 1.0 r) w) (/ (- 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))) / (((1.0 / r) / w) * ((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))) / (((1.0d0 / r) / w) * ((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))) / (((1.0 / r) / w) * ((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))) / (((1.0 / r) / w) * ((1.0 - v) / (r * w)))))) + -4.5

Julia

function code(v, w, r)
	return Float64(Float64(3.0 + Float64(Float64(2.0 / Float64(r * r)) - Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) / Float64(Float64(Float64(1.0 / r) / w) * Float64(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))) / (((1.0 / r) / w) * ((1.0 - v) / (r * w)))))) + -4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(3.0 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision] * N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 89.9%

   \[\leadsto \color{blue}{\left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right)\right) + -4.5} \]
4. Step-by-step derivation

   1. *-un-lft-identity 89.9%

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

   2. associate-*r* 83.5%

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

   3. swap-sqr 99.7%

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

   4. times-frac 99.7%

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

5. Applied egg-rr 99.7%

   \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\color{blue}{\frac{1}{r \cdot w} \cdot \frac{1 - v}{r \cdot w}}}\right)\right) + -4.5 \]
6. Step-by-step derivation

   1. inv-pow 99.7%

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

   2. unpow-prod-down 99.8%

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

   3. inv-pow 99.8%

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

7. Applied egg-rr 99.8%

   \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\color{blue}{\left(\frac{1}{r} \cdot {w}^{-1}\right)} \cdot \frac{1 - v}{r \cdot w}}\right)\right) + -4.5 \]
8. Step-by-step derivation

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

   3. unpow-1 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in w around 0 99.7%

    \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\color{blue}{\frac{1}{r \cdot w}} \cdot \frac{1 - v}{r \cdot w}}\right)\right) + -4.5 \]
11. Step-by-step derivation

    1. associate-/r* 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\\ \mathbf{if}\;v \leq -0.3:\\ \;\;\;\;\left(t_0 + t_1\right) + -1.5\\ \mathbf{elif}\;v \leq 1.9 \cdot 10^{-32}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{\frac{1}{r}}{w}}{r \cdot w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_1 + \frac{\frac{2}{r}}{r}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* (* (* r w) (* r w)) -0.25)))
   (if (<= v -0.3)
     (+ (+ t_0 t_1) -1.5)
     (if (<= v 1.9e-32)
       (+
        -4.5
        (+
         3.0
         (- t_0 (/ (* 0.125 (+ 3.0 (* -2.0 v))) (/ (/ (/ 1.0 r) w) (* r w))))))
       (+ -1.5 (+ t_1 (/ (/ 2.0 r) r)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((r * w) * (r * w)) * -0.25;
	double tmp;
	if (v <= -0.3) {
		tmp = (t_0 + t_1) + -1.5;
	} else if (v <= 1.9e-32) {
		tmp = -4.5 + (3.0 + (t_0 - ((0.125 * (3.0 + (-2.0 * v))) / (((1.0 / r) / w) / (r * w)))));
	} else {
		tmp = -1.5 + (t_1 + ((2.0 / r) / r));
	}
	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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = ((r * w) * (r * w)) * (-0.25d0)
    if (v <= (-0.3d0)) then
        tmp = (t_0 + t_1) + (-1.5d0)
    else if (v <= 1.9d-32) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) / (((1.0d0 / r) / w) / (r * w)))))
    else
        tmp = (-1.5d0) + (t_1 + ((2.0d0 / r) / r))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((r * w) * (r * w)) * -0.25;
	double tmp;
	if (v <= -0.3) {
		tmp = (t_0 + t_1) + -1.5;
	} else if (v <= 1.9e-32) {
		tmp = -4.5 + (3.0 + (t_0 - ((0.125 * (3.0 + (-2.0 * v))) / (((1.0 / r) / w) / (r * w)))));
	} else {
		tmp = -1.5 + (t_1 + ((2.0 / r) / r));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((r * w) * (r * w)) * -0.25
	tmp = 0
	if v <= -0.3:
		tmp = (t_0 + t_1) + -1.5
	elif v <= 1.9e-32:
		tmp = -4.5 + (3.0 + (t_0 - ((0.125 * (3.0 + (-2.0 * v))) / (((1.0 / r) / w) / (r * w)))))
	else:
		tmp = -1.5 + (t_1 + ((2.0 / r) / r))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25)
	tmp = 0.0
	if (v <= -0.3)
		tmp = Float64(Float64(t_0 + t_1) + -1.5);
	elseif (v <= 1.9e-32)
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) / Float64(Float64(Float64(1.0 / r) / w) / Float64(r * w))))));
	else
		tmp = Float64(-1.5 + Float64(t_1 + Float64(Float64(2.0 / r) / r)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((r * w) * (r * w)) * -0.25;
	tmp = 0.0;
	if (v <= -0.3)
		tmp = (t_0 + t_1) + -1.5;
	elseif (v <= 1.9e-32)
		tmp = -4.5 + (3.0 + (t_0 - ((0.125 * (3.0 + (-2.0 * v))) / (((1.0 / r) / w) / (r * w)))));
	else
		tmp = -1.5 + (t_1 + ((2.0 / r) / r));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]}, If[LessEqual[v, -0.3], N[(N[(t$95$0 + t$95$1), $MachinePrecision] + -1.5), $MachinePrecision], If[LessEqual[v, 1.9e-32], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$1 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -0.299999999999999989

   1. Initial program 78.1%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in v around inf 83.0%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -1.5 \]
   5. Step-by-step derivation

      1. *-commutative 83.0%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot -0.25}\right) + -1.5 \]

      2. unpow2 83.0%

         \[\leadsto \left(\frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}\right) \cdot -0.25\right) + -1.5 \]

      3. unpow2 83.0%

         \[\leadsto \left(\frac{2}{r \cdot r} + \left(\left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot w\right)}\right) \cdot -0.25\right) + -1.5 \]

      4. swap-sqr 98.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]

      5. unpow2 98.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot -0.25\right) + -1.5 \]

   6. Simplified 98.9%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{{\left(r \cdot w\right)}^{2} \cdot -0.25}\right) + -1.5 \]
   7. Step-by-step derivation

      1. unpow2 98.9%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]

   8. Applied egg-rr 98.9%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]

   if -0.299999999999999989 < v < 1.90000000000000004e-32

   1. Initial program 90.7%

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

   3. Simplified 90.6%

      \[\leadsto \color{blue}{\left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right)\right) + -4.5} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 90.6%

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

      2. associate-*r* 84.0%

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

      3. swap-sqr 99.7%

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

      4. times-frac 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\color{blue}{\frac{1}{r \cdot w} \cdot \frac{1 - v}{r \cdot w}}}\right)\right) + -4.5 \]
   6. Step-by-step derivation

      1. inv-pow 99.7%

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

      2. unpow-prod-down 99.8%

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

      3. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\color{blue}{\left(\frac{1}{r} \cdot {w}^{-1}\right)} \cdot \frac{1 - v}{r \cdot w}}\right)\right) + -4.5 \]
   8. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in v around 0 99.8%

       \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1}{w}}{r} \cdot \color{blue}{\frac{1}{r \cdot w}}}\right)\right) + -4.5 \]
   11. Step-by-step derivation

       1. un-div-inv 99.8%

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

       2. associate-/l/ 99.7%

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

       3. associate-/r* 99.8%

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

   12. Applied egg-rr 99.8%

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

   if 1.90000000000000004e-32 < v

   1. Initial program 79.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in v around inf 83.2%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -1.5 \]
   5. Step-by-step derivation

      1. *-commutative 83.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot -0.25}\right) + -1.5 \]

      2. unpow2 83.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}\right) \cdot -0.25\right) + -1.5 \]

      3. unpow2 83.2%

         \[\leadsto \left(\frac{2}{r \cdot r} + \left(\left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot w\right)}\right) \cdot -0.25\right) + -1.5 \]

      4. swap-sqr 98.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]

      5. unpow2 98.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot -0.25\right) + -1.5 \]

   6. Simplified 98.8%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{{\left(r \cdot w\right)}^{2} \cdot -0.25}\right) + -1.5 \]
   7. Step-by-step derivation

      1. unpow2 98.8%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]

   8. Applied egg-rr 98.8%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]
   9. Step-by-step derivation

      1. associate-/r* 98.8%

         \[\leadsto \left(\color{blue}{\frac{\frac{2}{r}}{r}} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]

      2. div-inv 98.8%

         \[\leadsto \left(\color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]

   10. Applied egg-rr 98.8%

       \[\leadsto \left(\color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]
   11. Step-by-step derivation

       1. associate-*r/ 98.8%

          \[\leadsto \left(\color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]

       2. *-rgt-identity 98.8%

          \[\leadsto \left(\frac{\color{blue}{\frac{2}{r}}}{r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]

   12. Simplified 98.8%

       \[\leadsto \left(\color{blue}{\frac{\frac{2}{r}}{r}} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -0.3:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5\\ \mathbf{elif}\;v \leq 1.9 \cdot 10^{-32}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{\frac{1}{r}}{w}}{r \cdot w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25 + \frac{\frac{2}{r}}{r}\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  -4.5
  (+
   3.0
   (-
    (/ 2.0 (* r r))
    (/
     (* 0.125 (+ 3.0 (* -2.0 v)))
     (* (/ (- 1.0 v) (* r w)) (/ 1.0 (* r w))))))))

C

double code(double v, double w, double r) {
	return -4.5 + (3.0 + ((2.0 / (r * r)) - ((0.125 * (3.0 + (-2.0 * v))) / (((1.0 - v) / (r * w)) * (1.0 / (r * w))))));
}

Fortran

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

Java

public static double code(double v, double w, double r) {
	return -4.5 + (3.0 + ((2.0 / (r * r)) - ((0.125 * (3.0 + (-2.0 * v))) / (((1.0 - v) / (r * w)) * (1.0 / (r * w))))));
}

Python

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

Julia

function code(v, w, r)
	return Float64(-4.5 + Float64(3.0 + Float64(Float64(2.0 / Float64(r * r)) - Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) / Float64(Float64(Float64(1.0 - v) / Float64(r * w)) * Float64(1.0 / Float64(r * w)))))))
end

MATLAB

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

Wolfram

code[v_, w_, r_] := N[(-4.5 + N[(3.0 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 89.9%

   \[\leadsto \color{blue}{\left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right)\right) + -4.5} \]
4. Step-by-step derivation

   1. *-un-lft-identity 89.9%

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

   2. associate-*r* 83.5%

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

   3. swap-sqr 99.7%

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

   4. times-frac 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{+284}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25 + \frac{\frac{2}{r}}{r}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= (* w w) 5e+284)
   (+
    -1.5
    (+
     (/ 2.0 (* r r))
     (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r (* r (* w w))))))
   (+ -1.5 (+ (* (* (* r w) (* r w)) -0.25) (/ (/ 2.0 r) r)))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((w * w) <= 5e+284) {
		tmp = -1.5 + ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = -1.5 + ((((r * w) * (r * w)) * -0.25) + ((2.0 / r) / r));
	}
	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
    real(8) :: tmp
    if ((w * w) <= 5d+284) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r * (r * (w * w)))))
    else
        tmp = (-1.5d0) + ((((r * w) * (r * w)) * (-0.25d0)) + ((2.0d0 / r) / r))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((w * w) <= 5e+284) {
		tmp = -1.5 + ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = -1.5 + ((((r * w) * (r * w)) * -0.25) + ((2.0 / r) / r));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (w * w) <= 5e+284:
		tmp = -1.5 + ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	else:
		tmp = -1.5 + ((((r * w) * (r * w)) * -0.25) + ((2.0 / r) / r))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (Float64(w * w) <= 5e+284)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))));
	else
		tmp = Float64(-1.5 + Float64(Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25) + Float64(Float64(2.0 / r) / r)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((w * w) <= 5e+284)
		tmp = -1.5 + ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	else
		tmp = -1.5 + ((((r * w) * (r * w)) * -0.25) + ((2.0 / r) / r));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(w * w), $MachinePrecision], 5e+284], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 4.9999999999999999e284

   1. Initial program 90.2%

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

   3. Simplified 97.5%

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

   if 4.9999999999999999e284 < (*.f64 w w)

   1. Initial program 66.2%

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

   3. Simplified 67.7%

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

   4. Taylor expanded in v around inf 67.6%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -1.5 \]
   5. Step-by-step derivation

      1. *-commutative 67.6%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot -0.25}\right) + -1.5 \]

      2. unpow2 67.6%

         \[\leadsto \left(\frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}\right) \cdot -0.25\right) + -1.5 \]

      3. unpow2 67.6%

         \[\leadsto \left(\frac{2}{r \cdot r} + \left(\left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot w\right)}\right) \cdot -0.25\right) + -1.5 \]

      4. swap-sqr 97.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]

      5. unpow2 97.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot -0.25\right) + -1.5 \]

   6. Simplified 97.4%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{{\left(r \cdot w\right)}^{2} \cdot -0.25}\right) + -1.5 \]
   7. Step-by-step derivation

      1. unpow2 97.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]

   8. Applied egg-rr 97.4%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]
   9. Step-by-step derivation

      1. associate-/r* 97.5%

         \[\leadsto \left(\color{blue}{\frac{\frac{2}{r}}{r}} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]

      2. div-inv 97.4%

         \[\leadsto \left(\color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]

   10. Applied egg-rr 97.4%

       \[\leadsto \left(\color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]
   11. Step-by-step derivation

       1. associate-*r/ 97.5%

          \[\leadsto \left(\color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]

       2. *-rgt-identity 97.5%

          \[\leadsto \left(\frac{\color{blue}{\frac{2}{r}}}{r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]

   12. Simplified 97.5%

       \[\leadsto \left(\color{blue}{\frac{\frac{2}{r}}{r}} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{+284}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25 + \frac{\frac{2}{r}}{r}\right)\\ \end{array} \]

Alternative 5: 93.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+ (+ (/ 2.0 (* r r)) (* (* (* r w) (* r w)) -0.25)) -1.5))

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.25)) + -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / (r * r)) + (((r * w) * (r * w)) * (-0.25d0))) + (-1.5d0)
end function

Java

public static double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.25)) + -1.5;
}

Python

def code(v, w, r):
	return ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.25)) + -1.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25)) + -1.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((2.0 / (r * r)) + (((r * w) * (r * w)) * -0.25)) + -1.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 90.0%

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

4. Taylor expanded in v around inf 78.9%

   \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{-0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -1.5 \]
5. Step-by-step derivation

   1. *-commutative 78.9%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot -0.25}\right) + -1.5 \]

   2. unpow2 78.9%

      \[\leadsto \left(\frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}\right) \cdot -0.25\right) + -1.5 \]

   3. unpow2 78.9%

      \[\leadsto \left(\frac{2}{r \cdot r} + \left(\left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot w\right)}\right) \cdot -0.25\right) + -1.5 \]

   4. swap-sqr 92.2%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]

   5. unpow2 92.2%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot -0.25\right) + -1.5 \]

6. Simplified 92.2%

   \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{{\left(r \cdot w\right)}^{2} \cdot -0.25}\right) + -1.5 \]
7. Step-by-step derivation

   1. unpow2 92.2%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]

8. Applied egg-rr 92.2%

   \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25\right) + -1.5 \]

9. Final simplification 92.2%

   \[\leadsto \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right) + -1.5 \]

Reproduce

herbie shell --seed 2023322 
(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))