Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 99.1%

Time: 16.9s

Alternatives: 11

Speedup: 1.5×

• 54.2% 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: 84.9% 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: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r} \cdot \frac{1}{r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{elif}\;v \leq 92000000:\\ \;\;\;\;\left(t\_0 + \left(r \cdot \left(w \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(w \cdot \left(r \cdot \frac{r \cdot w}{v}\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= v -1.55e+159)
     (- (- (+ 3.0 (* (/ 2.0 r) (/ 1.0 r))) (* (* (* r w) (* r w)) 0.25)) 4.5)
     (if (<= v 92000000.0)
       (-
        (+ t_0 (* (* r (* w (+ (* v -0.25) 0.375))) (* w (/ r (+ v -1.0)))))
        4.5)
       (+
        t_0
        (- (* (* 0.125 (+ 3.0 (* v -2.0))) (* w (* r (/ (* r w) v)))) 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= -1.55e+159) {
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (((r * w) * (r * w)) * 0.25)) - 4.5;
	} else if (v <= 92000000.0) {
		tmp = (t_0 + ((r * (w * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = t_0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * ((r * w) / v)))) - 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 = 3.0d0 + (2.0d0 / (r * r))
    if (v <= (-1.55d+159)) then
        tmp = ((3.0d0 + ((2.0d0 / r) * (1.0d0 / r))) - (((r * w) * (r * w)) * 0.25d0)) - 4.5d0
    else if (v <= 92000000.0d0) then
        tmp = (t_0 + ((r * (w * ((v * (-0.25d0)) + 0.375d0))) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    else
        tmp = t_0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * (w * (r * ((r * w) / v)))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= -1.55e+159) {
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (((r * w) * (r * w)) * 0.25)) - 4.5;
	} else if (v <= 92000000.0) {
		tmp = (t_0 + ((r * (w * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = t_0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * ((r * w) / v)))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if v <= -1.55e+159:
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (((r * w) * (r * w)) * 0.25)) - 4.5
	elif v <= 92000000.0:
		tmp = (t_0 + ((r * (w * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5
	else:
		tmp = t_0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * ((r * w) / v)))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (v <= -1.55e+159)
		tmp = Float64(Float64(Float64(3.0 + Float64(Float64(2.0 / r) * Float64(1.0 / r))) - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)) - 4.5);
	elseif (v <= 92000000.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(r * Float64(w * Float64(Float64(v * -0.25) + 0.375))) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	else
		tmp = Float64(t_0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(w * Float64(r * Float64(Float64(r * w) / v)))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if (v <= -1.55e+159)
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (((r * w) * (r * w)) * 0.25)) - 4.5;
	elseif (v <= 92000000.0)
		tmp = (t_0 + ((r * (w * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	else
		tmp = t_0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * ((r * w) / v)))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1.55e+159], N[(N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[v, 92000000.0], N[(N[(t$95$0 + N[(N[(r * N[(w * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r * N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.5499999999999999e159

   1. Initial program 75.6%

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

      1. associate-/l* 83.5%

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

      2. *-commutative 83.5%

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

      3. *-commutative 83.5%

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

      4. *-commutative 83.5%

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

      5. div-inv 83.5%

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

      6. associate-*l* 83.5%

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

      7. associate-*r* 78.8%

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

      8. *-commutative 78.8%

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

      9. pow2 78.8%

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

      10. pow2 78.8%

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

      11. pow-prod-down 95.5%

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

      12. cancel-sign-sub-inv 95.5%

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

      13. metadata-eval 95.5%

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

      14. +-commutative 95.5%

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

      15. distribute-rgt-in 95.5%

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

      16. *-commutative 95.5%

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

      17. associate-*l* 99.7%

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

      18. metadata-eval 99.7%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(w \cdot r\right)}^{2} \cdot \left(\frac{1}{1 - v} \cdot \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right)\right)\right) - 4.5 \]

      19. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 99.6%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.7%

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

   9. Applied egg-rr 99.7%

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

   if -1.5499999999999999e159 < v < 9.2e7

   1. Initial program 84.4%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in r around 0 99.9%

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

      1. associate-/l* 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 9.2e7 < v

   1. Initial program 79.6%

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

   3. Simplified 90.1%

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

      1. associate-/l* 90.1%

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

      2. *-commutative 90.1%

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

      3. associate-*r/ 90.2%

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

      4. *-commutative 90.2%

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

      5. associate-*l* 98.5%

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

      6. associate-*l* 99.8%

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

      7. associate-*r/ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. mul-1-neg 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r} \cdot \frac{1}{r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{elif}\;v \leq 92000000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(w \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(w \cdot \left(r \cdot \frac{r \cdot w}{v}\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\\ t_1 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6.5 \cdot 10^{+158}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r} \cdot \frac{1}{r}\right) - t\_0\right) - 4.5\\ \mathbf{elif}\;v \leq 15000000:\\ \;\;\;\;\left(t\_1 + \left(r \cdot \left(w \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - t\_0\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* (* r w) (* r w)) 0.25)) (t_1 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= v -6.5e+158)
     (- (- (+ 3.0 (* (/ 2.0 r) (/ 1.0 r))) t_0) 4.5)
     (if (<= v 15000000.0)
       (-
        (+ t_1 (* (* r (* w (+ (* v -0.25) 0.375))) (* w (/ r (+ v -1.0)))))
        4.5)
       (- (- t_1 t_0) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = ((r * w) * (r * w)) * 0.25;
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= -6.5e+158) {
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - t_0) - 4.5;
	} else if (v <= 15000000.0) {
		tmp = (t_1 + ((r * (w * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = (t_1 - t_0) - 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) :: t_1
    real(8) :: tmp
    t_0 = ((r * w) * (r * w)) * 0.25d0
    t_1 = 3.0d0 + (2.0d0 / (r * r))
    if (v <= (-6.5d+158)) then
        tmp = ((3.0d0 + ((2.0d0 / r) * (1.0d0 / r))) - t_0) - 4.5d0
    else if (v <= 15000000.0d0) then
        tmp = (t_1 + ((r * (w * ((v * (-0.25d0)) + 0.375d0))) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    else
        tmp = (t_1 - t_0) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = ((r * w) * (r * w)) * 0.25;
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= -6.5e+158) {
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - t_0) - 4.5;
	} else if (v <= 15000000.0) {
		tmp = (t_1 + ((r * (w * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = (t_1 - t_0) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((r * w) * (r * w)) * 0.25
	t_1 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if v <= -6.5e+158:
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - t_0) - 4.5
	elif v <= 15000000.0:
		tmp = (t_1 + ((r * (w * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5
	else:
		tmp = (t_1 - t_0) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)
	t_1 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (v <= -6.5e+158)
		tmp = Float64(Float64(Float64(3.0 + Float64(Float64(2.0 / r) * Float64(1.0 / r))) - t_0) - 4.5);
	elseif (v <= 15000000.0)
		tmp = Float64(Float64(t_1 + Float64(Float64(r * Float64(w * Float64(Float64(v * -0.25) + 0.375))) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	else
		tmp = Float64(Float64(t_1 - t_0) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((r * w) * (r * w)) * 0.25;
	t_1 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if (v <= -6.5e+158)
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - t_0) - 4.5;
	elseif (v <= 15000000.0)
		tmp = (t_1 + ((r * (w * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	else
		tmp = (t_1 - t_0) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -6.5e+158], N[(N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[v, 15000000.0], N[(N[(t$95$1 + N[(N[(r * N[(w * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - t$95$0), $MachinePrecision] - 4.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -6.5000000000000001e158

   1. Initial program 75.6%

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

      1. associate-/l* 83.5%

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

      2. *-commutative 83.5%

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

      3. *-commutative 83.5%

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

      4. *-commutative 83.5%

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

      5. div-inv 83.5%

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

      6. associate-*l* 83.5%

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

      7. associate-*r* 78.8%

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

      8. *-commutative 78.8%

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

      9. pow2 78.8%

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

      10. pow2 78.8%

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

      11. pow-prod-down 95.5%

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

      12. cancel-sign-sub-inv 95.5%

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

      13. metadata-eval 95.5%

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

      14. +-commutative 95.5%

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

      15. distribute-rgt-in 95.5%

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

      16. *-commutative 95.5%

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

      17. associate-*l* 99.7%

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

      18. metadata-eval 99.7%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(w \cdot r\right)}^{2} \cdot \left(\frac{1}{1 - v} \cdot \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right)\right)\right) - 4.5 \]

      19. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 99.6%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.7%

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

   9. Applied egg-rr 99.7%

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

   if -6.5000000000000001e158 < v < 1.5e7

   1. Initial program 84.4%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in r around 0 99.9%

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

      1. associate-/l* 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 1.5e7 < v

   1. Initial program 79.6%

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

      1. associate-/l* 90.1%

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

      2. *-commutative 90.1%

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

      3. *-commutative 90.1%

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

      4. *-commutative 90.1%

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

      5. div-inv 90.2%

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

      6. associate-*l* 90.1%

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

      7. associate-*r* 83.9%

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

      8. *-commutative 83.9%

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

      9. pow2 83.9%

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

      10. pow2 83.9%

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

      11. pow-prod-down 99.7%

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

      12. cancel-sign-sub-inv 99.7%

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

      13. metadata-eval 99.7%

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

      14. +-commutative 99.7%

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

      15. distribute-rgt-in 99.7%

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

      16. *-commutative 99.7%

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

      17. associate-*l* 99.7%

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

      18. metadata-eval 99.7%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(w \cdot r\right)}^{2} \cdot \left(\frac{1}{1 - v} \cdot \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right)\right)\right) - 4.5 \]

      19. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 99.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{0.25}\right) - 4.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6.5 \cdot 10^{+158}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r} \cdot \frac{1}{r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{elif}\;v \leq 15000000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(w \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -23.5 \lor \neg \left(v \leq 6.6 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\left(3 + \frac{2}{r} \cdot \frac{1}{r}\right) - t\_0 \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - t\_0 \cdot \left(0.375 + v \cdot 0.125\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))))
   (if (or (<= v -23.5) (not (<= v 6.6e-10)))
     (- (- (+ 3.0 (* (/ 2.0 r) (/ 1.0 r))) (* t_0 0.25)) 4.5)
     (- (- (+ 3.0 (/ 2.0 (* r r))) (* t_0 (+ 0.375 (* v 0.125)))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if ((v <= -23.5) || !(v <= 6.6e-10)) {
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (t_0 * 0.25)) - 4.5;
	} else {
		tmp = ((3.0 + (2.0 / (r * r))) - (t_0 * (0.375 + (v * 0.125)))) - 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 = (r * w) * (r * w)
    if ((v <= (-23.5d0)) .or. (.not. (v <= 6.6d-10))) then
        tmp = ((3.0d0 + ((2.0d0 / r) * (1.0d0 / r))) - (t_0 * 0.25d0)) - 4.5d0
    else
        tmp = ((3.0d0 + (2.0d0 / (r * r))) - (t_0 * (0.375d0 + (v * 0.125d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if ((v <= -23.5) || !(v <= 6.6e-10)) {
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (t_0 * 0.25)) - 4.5;
	} else {
		tmp = ((3.0 + (2.0 / (r * r))) - (t_0 * (0.375 + (v * 0.125)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	tmp = 0
	if (v <= -23.5) or not (v <= 6.6e-10):
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (t_0 * 0.25)) - 4.5
	else:
		tmp = ((3.0 + (2.0 / (r * r))) - (t_0 * (0.375 + (v * 0.125)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	tmp = 0.0
	if ((v <= -23.5) || !(v <= 6.6e-10))
		tmp = Float64(Float64(Float64(3.0 + Float64(Float64(2.0 / r) * Float64(1.0 / r))) - Float64(t_0 * 0.25)) - 4.5);
	else
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(t_0 * Float64(0.375 + Float64(v * 0.125)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	tmp = 0.0;
	if ((v <= -23.5) || ~((v <= 6.6e-10)))
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (t_0 * 0.25)) - 4.5;
	else
		tmp = ((3.0 + (2.0 / (r * r))) - (t_0 * (0.375 + (v * 0.125)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -23.5], N[Not[LessEqual[v, 6.6e-10]], $MachinePrecision]], N[(N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(0.375 + N[(v * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -23.5 or 6.6e-10 < v

   1. Initial program 79.2%

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

      1. associate-/l* 85.9%

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

      2. *-commutative 85.9%

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

      3. *-commutative 85.9%

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

      4. *-commutative 85.9%

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

      5. div-inv 85.9%

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

      6. associate-*l* 85.9%

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

      7. associate-*r* 80.2%

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

      8. *-commutative 80.2%

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

      9. pow2 80.2%

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

      10. pow2 80.2%

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

      11. pow-prod-down 98.9%

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

      12. cancel-sign-sub-inv 98.9%

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

      13. metadata-eval 98.9%

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

      14. +-commutative 98.9%

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

      15. distribute-rgt-in 98.9%

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

      16. *-commutative 98.9%

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

      17. associate-*l* 99.7%

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

      18. metadata-eval 99.7%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(w \cdot r\right)}^{2} \cdot \left(\frac{1}{1 - v} \cdot \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right)\right)\right) - 4.5 \]

      19. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 98.6%

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

      1. associate-/r* 98.7%

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

      2. div-inv 98.7%

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

   9. Applied egg-rr 98.7%

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

   if -23.5 < v < 6.6e-10

   1. Initial program 85.4%

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

      1. associate-/l* 85.4%

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

      2. *-commutative 85.4%

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

      3. *-commutative 85.4%

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

      4. *-commutative 85.4%

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

      5. div-inv 85.4%

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

      6. associate-*l* 85.4%

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

      7. associate-*r* 80.1%

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

      8. *-commutative 80.1%

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

      9. pow2 80.1%

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

      10. pow2 80.1%

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

      11. pow-prod-down 99.8%

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

      12. cancel-sign-sub-inv 99.8%

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

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

      15. distribute-rgt-in 99.8%

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

      16. *-commutative 99.8%

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

      17. associate-*l* 99.8%

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

      18. metadata-eval 99.8%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(w \cdot r\right)}^{2} \cdot \left(\frac{1}{1 - v} \cdot \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right)\right)\right) - 4.5 \]

      19. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -23.5 \lor \neg \left(v \leq 6.6 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\left(3 + \frac{2}{r} \cdot \frac{1}{r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.375 + v \cdot 0.125\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -23.5 \lor \neg \left(v \leq 7.8 \cdot 10^{-10}\right):\\ \;\;\;\;\left(t\_1 + t\_0 \cdot \left(\frac{0.125}{v} - 0.25\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - t\_0 \cdot \left(0.375 + v \cdot 0.125\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (+ 3.0 (/ 2.0 (* r r)))))
   (if (or (<= v -23.5) (not (<= v 7.8e-10)))
     (- (+ t_1 (* t_0 (- (/ 0.125 v) 0.25))) 4.5)
     (- (- t_1 (* t_0 (+ 0.375 (* v 0.125)))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if ((v <= -23.5) || !(v <= 7.8e-10)) {
		tmp = (t_1 + (t_0 * ((0.125 / v) - 0.25))) - 4.5;
	} else {
		tmp = (t_1 - (t_0 * (0.375 + (v * 0.125)))) - 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) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = 3.0d0 + (2.0d0 / (r * r))
    if ((v <= (-23.5d0)) .or. (.not. (v <= 7.8d-10))) then
        tmp = (t_1 + (t_0 * ((0.125d0 / v) - 0.25d0))) - 4.5d0
    else
        tmp = (t_1 - (t_0 * (0.375d0 + (v * 0.125d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if ((v <= -23.5) || !(v <= 7.8e-10)) {
		tmp = (t_1 + (t_0 * ((0.125 / v) - 0.25))) - 4.5;
	} else {
		tmp = (t_1 - (t_0 * (0.375 + (v * 0.125)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if (v <= -23.5) or not (v <= 7.8e-10):
		tmp = (t_1 + (t_0 * ((0.125 / v) - 0.25))) - 4.5
	else:
		tmp = (t_1 - (t_0 * (0.375 + (v * 0.125)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if ((v <= -23.5) || !(v <= 7.8e-10))
		tmp = Float64(Float64(t_1 + Float64(t_0 * Float64(Float64(0.125 / v) - 0.25))) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(t_0 * Float64(0.375 + Float64(v * 0.125)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if ((v <= -23.5) || ~((v <= 7.8e-10)))
		tmp = (t_1 + (t_0 * ((0.125 / v) - 0.25))) - 4.5;
	else
		tmp = (t_1 - (t_0 * (0.375 + (v * 0.125)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -23.5], N[Not[LessEqual[v, 7.8e-10]], $MachinePrecision]], N[(N[(t$95$1 + N[(t$95$0 * N[(N[(0.125 / v), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(t$95$0 * N[(0.375 + N[(v * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -23.5 or 7.7999999999999999e-10 < v

   1. Initial program 79.2%

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

      1. associate-/l* 85.9%

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

      2. *-commutative 85.9%

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

      3. *-commutative 85.9%

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

      4. *-commutative 85.9%

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

      5. div-inv 85.9%

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

      6. associate-*l* 85.9%

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

      7. associate-*r* 80.2%

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

      8. *-commutative 80.2%

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

      9. pow2 80.2%

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

      10. pow2 80.2%

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

      11. pow-prod-down 98.9%

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

      12. cancel-sign-sub-inv 98.9%

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

      13. metadata-eval 98.9%

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

      14. +-commutative 98.9%

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

      15. distribute-rgt-in 98.9%

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

      16. *-commutative 98.9%

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

      17. associate-*l* 99.7%

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

      18. metadata-eval 99.7%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(w \cdot r\right)}^{2} \cdot \left(\frac{1}{1 - v} \cdot \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right)\right)\right) - 4.5 \]

      19. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 98.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\left(0.25 - 0.125 \cdot \frac{1}{v}\right)}\right) - 4.5 \]
   8. Step-by-step derivation

      1. associate-*r/ 98.8%

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \left(0.25 - \color{blue}{\frac{0.125 \cdot 1}{v}}\right)\right) - 4.5 \]

      2. metadata-eval 98.8%

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \left(0.25 - \frac{\color{blue}{0.125}}{v}\right)\right) - 4.5 \]

   9. Simplified 98.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\left(0.25 - \frac{0.125}{v}\right)}\right) - 4.5 \]

   if -23.5 < v < 7.7999999999999999e-10

   1. Initial program 85.4%

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

      1. associate-/l* 85.4%

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

      2. *-commutative 85.4%

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

      3. *-commutative 85.4%

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

      4. *-commutative 85.4%

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

      5. div-inv 85.4%

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

      6. associate-*l* 85.4%

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

      7. associate-*r* 80.1%

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

      8. *-commutative 80.1%

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

      9. pow2 80.1%

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

      10. pow2 80.1%

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

      11. pow-prod-down 99.8%

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

      12. cancel-sign-sub-inv 99.8%

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

      13. metadata-eval 99.8%

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

      14. +-commutative 99.8%

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

      15. distribute-rgt-in 99.8%

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

      16. *-commutative 99.8%

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

      17. associate-*l* 99.8%

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

      18. metadata-eval 99.8%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(w \cdot r\right)}^{2} \cdot \left(\frac{1}{1 - v} \cdot \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right)\right)\right) - 4.5 \]

      19. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -23.5 \lor \neg \left(v \leq 7.8 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.375 + v \cdot 0.125\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -23.5 \lor \neg \left(v \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\left(3 + \frac{2}{r} \cdot \frac{1}{r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -23.5) (not (<= v 5e-10)))
   (- (- (+ 3.0 (* (/ 2.0 r) (/ 1.0 r))) (* (* (* r w) (* r w)) 0.25)) 4.5)
   (- (- (+ 3.0 (/ 2.0 (* r r))) (* (* r w) (* r (* w 0.375)))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -23.5) || !(v <= 5e-10)) {
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (((r * w) * (r * w)) * 0.25)) - 4.5;
	} else {
		tmp = ((3.0 + (2.0 / (r * r))) - ((r * w) * (r * (w * 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) :: tmp
    if ((v <= (-23.5d0)) .or. (.not. (v <= 5d-10))) then
        tmp = ((3.0d0 + ((2.0d0 / r) * (1.0d0 / r))) - (((r * w) * (r * w)) * 0.25d0)) - 4.5d0
    else
        tmp = ((3.0d0 + (2.0d0 / (r * r))) - ((r * w) * (r * (w * 0.375d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -23.5) || !(v <= 5e-10)) {
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (((r * w) * (r * w)) * 0.25)) - 4.5;
	} else {
		tmp = ((3.0 + (2.0 / (r * r))) - ((r * w) * (r * (w * 0.375)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -23.5) or not (v <= 5e-10):
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (((r * w) * (r * w)) * 0.25)) - 4.5
	else:
		tmp = ((3.0 + (2.0 / (r * r))) - ((r * w) * (r * (w * 0.375)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -23.5) || !(v <= 5e-10))
		tmp = Float64(Float64(Float64(3.0 + Float64(Float64(2.0 / r) * Float64(1.0 / r))) - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)) - 4.5);
	else
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(r * w) * Float64(r * Float64(w * 0.375)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -23.5) || ~((v <= 5e-10)))
		tmp = ((3.0 + ((2.0 / r) * (1.0 / r))) - (((r * w) * (r * w)) * 0.25)) - 4.5;
	else
		tmp = ((3.0 + (2.0 / (r * r))) - ((r * w) * (r * (w * 0.375)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -23.5], N[Not[LessEqual[v, 5e-10]], $MachinePrecision]], N[(N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -23.5 or 5.00000000000000031e-10 < v

   1. Initial program 79.2%

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

      1. associate-/l* 85.9%

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

      2. *-commutative 85.9%

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

      3. *-commutative 85.9%

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

      4. *-commutative 85.9%

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

      5. div-inv 85.9%

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

      6. associate-*l* 85.9%

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

      7. associate-*r* 80.2%

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

      8. *-commutative 80.2%

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

      9. pow2 80.2%

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

      10. pow2 80.2%

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

      11. pow-prod-down 98.9%

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

      12. cancel-sign-sub-inv 98.9%

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

      13. metadata-eval 98.9%

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

      14. +-commutative 98.9%

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

      15. distribute-rgt-in 98.9%

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

      16. *-commutative 98.9%

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

      17. associate-*l* 99.7%

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

      18. metadata-eval 99.7%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(w \cdot r\right)}^{2} \cdot \left(\frac{1}{1 - v} \cdot \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right)\right)\right) - 4.5 \]

      19. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 98.6%

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

      1. associate-/r* 98.7%

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

      2. div-inv 98.7%

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

   9. Applied egg-rr 98.7%

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

   if -23.5 < v < 5.00000000000000031e-10

   1. Initial program 85.4%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in r around 0 99.9%

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

   6. Taylor expanded in v around 0 99.6%

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

   7. Taylor expanded in v around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -23.5 \lor \neg \left(v \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\left(3 + \frac{2}{r} \cdot \frac{1}{r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -23.5 \lor \neg \left(v \leq 7 \cdot 10^{-10}\right):\\ \;\;\;\;\left(t\_0 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (or (<= v -23.5) (not (<= v 7e-10)))
     (- (- t_0 (* (* (* r w) (* r w)) 0.25)) 4.5)
     (- (- t_0 (* (* r w) (* r (* w 0.375)))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if ((v <= -23.5) || !(v <= 7e-10)) {
		tmp = (t_0 - (((r * w) * (r * w)) * 0.25)) - 4.5;
	} else {
		tmp = (t_0 - ((r * w) * (r * (w * 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 = 3.0d0 + (2.0d0 / (r * r))
    if ((v <= (-23.5d0)) .or. (.not. (v <= 7d-10))) then
        tmp = (t_0 - (((r * w) * (r * w)) * 0.25d0)) - 4.5d0
    else
        tmp = (t_0 - ((r * w) * (r * (w * 0.375d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if ((v <= -23.5) || !(v <= 7e-10)) {
		tmp = (t_0 - (((r * w) * (r * w)) * 0.25)) - 4.5;
	} else {
		tmp = (t_0 - ((r * w) * (r * (w * 0.375)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if (v <= -23.5) or not (v <= 7e-10):
		tmp = (t_0 - (((r * w) * (r * w)) * 0.25)) - 4.5
	else:
		tmp = (t_0 - ((r * w) * (r * (w * 0.375)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if ((v <= -23.5) || !(v <= 7e-10))
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(r * w) * Float64(r * Float64(w * 0.375)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if ((v <= -23.5) || ~((v <= 7e-10)))
		tmp = (t_0 - (((r * w) * (r * w)) * 0.25)) - 4.5;
	else
		tmp = (t_0 - ((r * w) * (r * (w * 0.375)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -23.5], N[Not[LessEqual[v, 7e-10]], $MachinePrecision]], N[(N[(t$95$0 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -23.5 or 6.99999999999999961e-10 < v

   1. Initial program 79.2%

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

      1. associate-/l* 85.9%

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

      2. *-commutative 85.9%

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

      3. *-commutative 85.9%

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

      4. *-commutative 85.9%

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

      5. div-inv 85.9%

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

      6. associate-*l* 85.9%

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

      7. associate-*r* 80.2%

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

      8. *-commutative 80.2%

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

      9. pow2 80.2%

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

      10. pow2 80.2%

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

      11. pow-prod-down 98.9%

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

      12. cancel-sign-sub-inv 98.9%

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

      13. metadata-eval 98.9%

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

      14. +-commutative 98.9%

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

      15. distribute-rgt-in 98.9%

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

      16. *-commutative 98.9%

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

      17. associate-*l* 99.7%

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

      18. metadata-eval 99.7%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(w \cdot r\right)}^{2} \cdot \left(\frac{1}{1 - v} \cdot \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right)\right)\right) - 4.5 \]

      19. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 98.6%

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

   if -23.5 < v < 6.99999999999999961e-10

   1. Initial program 85.4%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in r around 0 99.9%

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

   6. Taylor expanded in v around 0 99.6%

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

   7. Taylor expanded in v around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -23.5 \lor \neg \left(v \leq 7 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+
   (+ 3.0 (/ 2.0 (* r r)))
   (* (* (* r w) (* r w)) (* (+ (* v -0.25) 0.375) (/ -1.0 (- 1.0 v)))))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) + (((r * w) * (r * w)) * (((v * -0.25) + 0.375) * (-1.0 / (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))) + (((r * w) * (r * w)) * (((v * (-0.25d0)) + 0.375d0) * ((-1.0d0) / (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))) + (((r * w) * (r * w)) * (((v * -0.25) + 0.375) * (-1.0 / (1.0 - v))))) - 4.5;
}

Python

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

MATLAB

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

Derivation

1. Initial program 82.4%

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

   1. associate-/l* 85.7%

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

   2. *-commutative 85.7%

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

   3. *-commutative 85.7%

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

   4. *-commutative 85.7%

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

   5. div-inv 85.7%

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

   6. associate-*l* 85.7%

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

   7. associate-*r* 80.2%

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

   8. *-commutative 80.2%

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

   9. pow2 80.2%

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

   10. pow2 80.2%

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

   11. pow-prod-down 99.3%

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

   12. cancel-sign-sub-inv 99.3%

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

   13. metadata-eval 99.3%

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

   14. +-commutative 99.3%

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

   15. distribute-rgt-in 99.3%

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

   16. *-commutative 99.3%

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

   17. associate-*l* 99.7%

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

   18. metadata-eval 99.7%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(w \cdot r\right)}^{2} \cdot \left(\frac{1}{1 - v} \cdot \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right)\right)\right) - 4.5 \]

   19. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 8: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (3.0 + (2.0 / (r * r))) + (((0.125 * (3.0 + (v * -2.0))) * (w * ((r * w) * (r / (v + -1.0))))) - 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 = (3.0d0 + (2.0d0 / (r * r))) + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * (w * ((r * w) * (r / (v + (-1.0d0)))))) - 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 + (v * -2.0))) * (w * ((r * w) * (r / (v + -1.0))))) - 4.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Simplified 85.7%

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

   1. associate-/l* 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-*r/ 85.3%

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

   4. associate-*l* 95.9%

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

   5. associate-*r* 99.0%

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

   6. add-sqr-sqrt 47.9%

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

   7. associate-*l* 47.9%

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

   8. add-sqr-sqrt 26.5%

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

   9. sqrt-prod 33.7%

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

   10. sqrt-prod 33.7%

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

   11. sqrt-prod 71.8%

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

   12. associate-*l* 71.8%

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

   13. *-commutative 71.8%

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

   14. associate-*l* 71.4%

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

6. Applied egg-rr 97.9%

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

7. Final simplification 97.9%

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

Alternative 9: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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))) - (4.5d0 - ((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * (w * (r * ((r * w) / (v + (-1.0d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Simplified 85.7%

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

   1. associate-/l* 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-*r/ 85.3%

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

   4. *-commutative 85.3%

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

   5. associate-*l* 95.9%

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

   6. associate-*l* 97.9%

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

   7. associate-*r/ 98.3%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 10: 93.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

4. Applied egg-rr 93.1%

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

5. Taylor expanded in v around 0 84.9%

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

6. Taylor expanded in v around 0 93.5%

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

7. Final simplification 93.5%

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

Alternative 11: 93.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

4. Applied egg-rr 93.1%

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

5. Taylor expanded in r around 0 93.8%

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

6. Taylor expanded in v around 0 79.0%

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

7. Taylor expanded in v around 0 93.5%

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

   1. *-commutative 93.5%

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

   2. associate-*l* 93.5%

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

9. Simplified 93.5%

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

10. Final simplification 93.5%

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

Reproduce

herbie shell --seed 2024130 
(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))