Rosa's TurbineBenchmark

Percentage Accurate: 84.4% → 99.8%

Time: 14.5s

Alternatives: 6

Speedup: 1.5×

• 53.3% 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: 84.4% 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: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 + \frac{0.375 + v \cdot -0.25}{\frac{v + -1}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (+ -1.5 (/ (+ 0.375 (* v -0.25)) (/ (+ v -1.0) (* (* r w) (* r w)))))))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) / ((v + -1.0) / ((r * w) * (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 = (2.0d0 / (r * r)) + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) / ((v + (-1.0d0)) / ((r * w) * (r * w)))))
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) / ((v + -1.0) / ((r * w) * (r * w)))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) / ((v + -1.0) / ((r * w) * (r * w)))))

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(Float64(v + -1.0) / Float64(Float64(r * w) * Float64(r * w))))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) / ((v + -1.0) / ((r * w) * (r * w)))));
end

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Simplified 89.2%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 89.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   2. *-commutative 89.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   3. +-commutative 89.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   4. metadata-eval 89.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + \color{blue}{\left(-2\right)} \cdot v\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   5. cancel-sign-sub-inv 89.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 - 2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   6. associate-*r/ 89.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

   7. *-commutative 89.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

   8. associate-/l* 89.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

   9. clear-num 89.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   10. un-div-inv 89.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   11. cancel-sign-sub-inv 89.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(3 + \left(-2\right) \cdot v\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. metadata-eval 89.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \left(3 + \color{blue}{-2} \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. distribute-rgt-in 89.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{3 \cdot 0.125 + \left(-2 \cdot v\right) \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   14. metadata-eval 89.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   15. *-commutative 89.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   16. associate-*l* 89.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   17. metadata-eval 89.9%

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

6. Applied egg-rr 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.375 + v \cdot -0.25}{\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}}\right) \]
7. Step-by-step derivation

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 + \frac{0.375 + v \cdot -0.25}{\frac{v + -1}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]
10. Add Preprocessing

Alternative 2: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ \mathbf{if}\;v \leq -5000000:\\ \;\;\;\;\left(t\_0 + \left(r \cdot w\right) \cdot \frac{w \cdot \left(r \cdot \left(v \cdot -0.25\right)\right)}{v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) 3.0)))
   (if (<= v -5000000.0)
     (- (+ t_0 (* (* r w) (/ (* w (* r (* v -0.25))) v))) 4.5)
     (- (- t_0 (* (* r w) (* w (* r 0.375)))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + 3.0;
	double tmp;
	if (v <= -5000000.0) {
		tmp = (t_0 + ((r * w) * ((w * (r * (v * -0.25))) / v))) - 4.5;
	} else {
		tmp = (t_0 - ((r * w) * (w * (r * 0.375)))) - 4.5;
	}
	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) :: tmp
    t_0 = (2.0d0 / (r * r)) + 3.0d0
    if (v <= (-5000000.0d0)) then
        tmp = (t_0 + ((r * w) * ((w * (r * (v * (-0.25d0)))) / v))) - 4.5d0
    else
        tmp = (t_0 - ((r * w) * (w * (r * 0.375d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + 3.0;
	double tmp;
	if (v <= -5000000.0) {
		tmp = (t_0 + ((r * w) * ((w * (r * (v * -0.25))) / v))) - 4.5;
	} else {
		tmp = (t_0 - ((r * w) * (w * (r * 0.375)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + 3.0
	tmp = 0
	if v <= -5000000.0:
		tmp = (t_0 + ((r * w) * ((w * (r * (v * -0.25))) / v))) - 4.5
	else:
		tmp = (t_0 - ((r * w) * (w * (r * 0.375)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + 3.0)
	tmp = 0.0
	if (v <= -5000000.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(r * w) * Float64(Float64(w * Float64(r * Float64(v * -0.25))) / v))) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + 3.0;
	tmp = 0.0;
	if (v <= -5000000.0)
		tmp = (t_0 + ((r * w) * ((w * (r * (v * -0.25))) / v))) - 4.5;
	else
		tmp = (t_0 - ((r * w) * (w * (r * 0.375)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[v, -5000000.0], N[(N[(t$95$0 + N[(N[(r * w), $MachinePrecision] * N[(N[(w * N[(r * N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -5e6

   1. Initial program 81.8%

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

      1. associate-/l* 88.1%

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

      2. cancel-sign-sub-inv 88.1%

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

      3. metadata-eval 88.1%

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

      4. +-commutative 88.1%

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

      5. *-commutative 88.1%

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

      6. fma-undefine 88.1%

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

      7. *-commutative 88.1%

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

      8. *-commutative 88.1%

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

      9. associate-/l* 88.1%

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

      10. *-commutative 88.1%

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

      11. associate-*r/ 88.1%

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

      12. associate-*r* 83.1%

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

      13. associate-*l* 93.2%

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

      14. associate-*r* 89.9%

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

   4. Applied egg-rr 89.9%

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

      1. *-commutative 89.9%

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

      2. associate-/r/ 89.9%

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

      3. *-commutative 89.9%

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

      4. *-commutative 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in v around inf 84.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{w}{1 - v} \cdot r\right) \cdot \color{blue}{\left(-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)\right)}\right) - 4.5 \]
   8. Step-by-step derivation

      1. associate-*r* 84.1%

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

      2. *-commutative 84.1%

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

      3. *-commutative 84.1%

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

      4. associate-*r* 89.3%

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

      5. *-commutative 89.3%

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

      6. associate-*r* 89.3%

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

      7. *-commutative 89.3%

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

      8. associate-*l* 84.1%

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

      9. *-commutative 84.1%

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

   9. Simplified 84.1%

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

   10. Taylor expanded in v around inf 85.7%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(-1 \cdot \frac{r \cdot w}{v}\right)} \cdot \left(r \cdot \left(w \cdot \left(v \cdot -0.25\right)\right)\right)\right) - 4.5 \]
   11. Step-by-step derivation

       1. associate-*r/ 85.7%

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

       2. *-commutative 85.7%

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

       3. neg-mul-1 85.7%

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

       4. distribute-rgt-neg-in 85.7%

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

   12. Simplified 85.7%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{w \cdot \left(-r\right)}{v}} \cdot \left(r \cdot \left(w \cdot \left(v \cdot -0.25\right)\right)\right)\right) - 4.5 \]
   13. Step-by-step derivation

       1. associate-*l/ 82.6%

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

       2. associate-*r* 91.2%

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

       3. *-commutative 91.2%

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

   14. Applied egg-rr 91.2%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\left(w \cdot \left(-r\right)\right) \cdot \left(\left(w \cdot r\right) \cdot \left(v \cdot -0.25\right)\right)}{v}}\right) - 4.5 \]
   15. Step-by-step derivation

       1. associate-/l* 94.3%

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

       2. associate-*l* 89.4%

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

   16. Simplified 89.4%

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

   if -5e6 < v

   1. Initial program 88.5%

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

      1. associate-/l* 90.4%

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

      2. cancel-sign-sub-inv 90.4%

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

      3. metadata-eval 90.4%

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

      4. +-commutative 90.4%

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

      5. *-commutative 90.4%

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

      6. fma-undefine 90.4%

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

      7. *-commutative 90.4%

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

      8. *-commutative 90.4%

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

      9. associate-/l* 89.5%

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

      10. *-commutative 89.5%

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

      11. associate-*r/ 89.5%

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

      12. associate-*r* 87.9%

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

      13. associate-*l* 94.5%

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

      14. associate-*r* 96.9%

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

   4. Applied egg-rr 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-/r/ 96.8%

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

      3. *-commutative 96.8%

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

      4. *-commutative 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in v around 0 85.3%

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

   8. Taylor expanded in v around 0 96.2%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5000000:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \left(r \cdot w\right) \cdot \frac{w \cdot \left(r \cdot \left(v \cdot -0.25\right)\right)}{v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ \mathbf{if}\;v \leq -1700000:\\ \;\;\;\;\left(t\_0 + \frac{\left(\left(v \cdot -0.25\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) 3.0)))
   (if (<= v -1700000.0)
     (- (+ t_0 (/ (* (* (* v -0.25) (* r w)) (* r w)) v)) 4.5)
     (- (- t_0 (* (* r w) (* w (* r 0.375)))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + 3.0;
	double tmp;
	if (v <= -1700000.0) {
		tmp = (t_0 + ((((v * -0.25) * (r * w)) * (r * w)) / v)) - 4.5;
	} else {
		tmp = (t_0 - ((r * w) * (w * (r * 0.375)))) - 4.5;
	}
	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) :: tmp
    t_0 = (2.0d0 / (r * r)) + 3.0d0
    if (v <= (-1700000.0d0)) then
        tmp = (t_0 + ((((v * (-0.25d0)) * (r * w)) * (r * w)) / v)) - 4.5d0
    else
        tmp = (t_0 - ((r * w) * (w * (r * 0.375d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + 3.0;
	double tmp;
	if (v <= -1700000.0) {
		tmp = (t_0 + ((((v * -0.25) * (r * w)) * (r * w)) / v)) - 4.5;
	} else {
		tmp = (t_0 - ((r * w) * (w * (r * 0.375)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + 3.0
	tmp = 0
	if v <= -1700000.0:
		tmp = (t_0 + ((((v * -0.25) * (r * w)) * (r * w)) / v)) - 4.5
	else:
		tmp = (t_0 - ((r * w) * (w * (r * 0.375)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + 3.0)
	tmp = 0.0
	if (v <= -1700000.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(Float64(v * -0.25) * Float64(r * w)) * Float64(r * w)) / v)) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + 3.0;
	tmp = 0.0;
	if (v <= -1700000.0)
		tmp = (t_0 + ((((v * -0.25) * (r * w)) * (r * w)) / v)) - 4.5;
	else
		tmp = (t_0 - ((r * w) * (w * (r * 0.375)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[v, -1700000.0], N[(N[(t$95$0 + N[(N[(N[(N[(v * -0.25), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.7e6

   1. Initial program 81.8%

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

      1. associate-/l* 88.1%

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

      2. cancel-sign-sub-inv 88.1%

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

      3. metadata-eval 88.1%

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

      4. +-commutative 88.1%

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

      5. *-commutative 88.1%

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

      6. fma-undefine 88.1%

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

      7. *-commutative 88.1%

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

      8. *-commutative 88.1%

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

      9. associate-/l* 88.1%

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

      10. *-commutative 88.1%

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

      11. associate-*r/ 88.1%

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

      12. associate-*r* 83.1%

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

      13. associate-*l* 93.2%

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

      14. associate-*r* 89.9%

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

   4. Applied egg-rr 89.9%

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

      1. *-commutative 89.9%

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

      2. associate-/r/ 89.9%

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

      3. *-commutative 89.9%

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

      4. *-commutative 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in v around inf 84.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{w}{1 - v} \cdot r\right) \cdot \color{blue}{\left(-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)\right)}\right) - 4.5 \]
   8. Step-by-step derivation

      1. associate-*r* 84.1%

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

      2. *-commutative 84.1%

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

      3. *-commutative 84.1%

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

      4. associate-*r* 89.3%

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

      5. *-commutative 89.3%

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

      6. associate-*r* 89.3%

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

      7. *-commutative 89.3%

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

      8. associate-*l* 84.1%

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

      9. *-commutative 84.1%

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

   9. Simplified 84.1%

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

   10. Taylor expanded in v around inf 85.7%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(-1 \cdot \frac{r \cdot w}{v}\right)} \cdot \left(r \cdot \left(w \cdot \left(v \cdot -0.25\right)\right)\right)\right) - 4.5 \]
   11. Step-by-step derivation

       1. associate-*r/ 85.7%

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

       2. *-commutative 85.7%

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

       3. neg-mul-1 85.7%

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

       4. distribute-rgt-neg-in 85.7%

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

   12. Simplified 85.7%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{w \cdot \left(-r\right)}{v}} \cdot \left(r \cdot \left(w \cdot \left(v \cdot -0.25\right)\right)\right)\right) - 4.5 \]
   13. Step-by-step derivation

       1. associate-*l/ 82.6%

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

       2. associate-*r* 91.2%

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

       3. *-commutative 91.2%

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

   14. Applied egg-rr 91.2%

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

   if -1.7e6 < v

   1. Initial program 88.5%

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

      1. associate-/l* 90.4%

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

      2. cancel-sign-sub-inv 90.4%

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

      3. metadata-eval 90.4%

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

      4. +-commutative 90.4%

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

      5. *-commutative 90.4%

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

      6. fma-undefine 90.4%

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

      7. *-commutative 90.4%

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

      8. *-commutative 90.4%

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

      9. associate-/l* 89.5%

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

      10. *-commutative 89.5%

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

      11. associate-*r/ 89.5%

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

      12. associate-*r* 87.9%

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

      13. associate-*l* 94.5%

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

      14. associate-*r* 96.9%

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

   4. Applied egg-rr 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-/r/ 96.8%

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

      3. *-commutative 96.8%

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

      4. *-commutative 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in v around 0 85.3%

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

   8. Taylor expanded in v around 0 96.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1700000:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(\left(v \cdot -0.25\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (- (- (+ (/ 2.0 (* r r)) 3.0) (* (* r w) (* 0.375 (* r w)))) 4.5))

C

double code(double v, double w, double r) {
	return (((2.0 / (r * r)) + 3.0) - ((r * w) * (0.375 * (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 = (((2.0d0 / (r * r)) + 3.0d0) - ((r * w) * (0.375d0 * (r * w)))) - 4.5d0
end function

Java

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

Python

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

Julia

function code(v, w, r)
	return Float64(Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(r * w) * Float64(0.375 * Float64(r * w)))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = (((2.0 / (r * r)) + 3.0) - ((r * w) * (0.375 * (r * w)))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 87.0%

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

   1. associate-/l* 89.8%

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

   2. cancel-sign-sub-inv 89.8%

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

   3. metadata-eval 89.8%

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

   4. +-commutative 89.8%

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

   5. *-commutative 89.8%

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

   6. fma-undefine 89.8%

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

   7. *-commutative 89.8%

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

   8. *-commutative 89.8%

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

   9. associate-/l* 89.1%

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

   10. *-commutative 89.1%

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

   11. associate-*r/ 89.1%

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

   12. associate-*r* 86.8%

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

   13. associate-*l* 94.2%

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

   14. associate-*r* 95.3%

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

4. Applied egg-rr 95.3%

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

   1. *-commutative 95.3%

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

   2. associate-/r/ 95.2%

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

   3. *-commutative 95.2%

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

   4. *-commutative 95.2%

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

6. Simplified 95.2%

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

7. Taylor expanded in v around 0 82.8%

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

8. Taylor expanded in v around 0 93.0%

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

9. Final simplification 93.0%

   \[\leadsto \left(\left(\frac{2}{r \cdot r} + 3\right) - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right) - 4.5 \]
10. Add Preprocessing

Alternative 5: 93.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (- (- (+ (/ 2.0 (* r r)) 3.0) (* (* r w) (* w (* r 0.375)))) 4.5))

C

double code(double v, double w, double r) {
	return (((2.0 / (r * r)) + 3.0) - ((r * w) * (w * (r * 0.375)))) - 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 = (((2.0d0 / (r * r)) + 3.0d0) - ((r * w) * (w * (r * 0.375d0)))) - 4.5d0
end function

Java

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

Python

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

Julia

function code(v, w, r)
	return Float64(Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = (((2.0 / (r * r)) + 3.0) - ((r * w) * (w * (r * 0.375)))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 87.0%

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

   1. associate-/l* 89.8%

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

   2. cancel-sign-sub-inv 89.8%

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

   3. metadata-eval 89.8%

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

   4. +-commutative 89.8%

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

   5. *-commutative 89.8%

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

   6. fma-undefine 89.8%

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

   7. *-commutative 89.8%

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

   8. *-commutative 89.8%

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

   9. associate-/l* 89.1%

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

   10. *-commutative 89.1%

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

   11. associate-*r/ 89.1%

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

   12. associate-*r* 86.8%

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

   13. associate-*l* 94.2%

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

   14. associate-*r* 95.3%

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

4. Applied egg-rr 95.3%

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

   1. *-commutative 95.3%

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

   2. associate-/r/ 95.2%

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

   3. *-commutative 95.2%

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

   4. *-commutative 95.2%

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

6. Simplified 95.2%

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

7. Taylor expanded in v around 0 82.8%

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

8. Taylor expanded in v around 0 93.0%

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

9. Final simplification 93.0%

   \[\leadsto \left(\left(\frac{2}{r \cdot r} + 3\right) - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right) - 4.5 \]
10. Add Preprocessing

Alternative 6: 57.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + 3.0) - 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 = ((2.0d0 / (r * r)) + 3.0d0) - 4.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Simplified 82.9%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 57.1%

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

6. Final simplification 57.1%

   \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - 4.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))