Rosa's TurbineBenchmark

Percentage Accurate: 84.2% → 99.3%

Time: 15.7s

Alternatives: 11

Speedup: 1.1×

• 53.0% 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: 84.2% 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: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 3 + t\_0\\ \mathbf{if}\;v \leq -4 \cdot 10^{+28}:\\ \;\;\;\;t\_1 + \left(\left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v}\right)\right) - 4.5\right)\\ \mathbf{elif}\;v \leq 3 \cdot 10^{+108}:\\ \;\;\;\;\left(t\_1 + \left(w \cdot \left(r \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot \frac{v}{1 - v}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ 3.0 t_0)))
   (if (<= v -4e+28)
     (+ t_1 (- (* (* v -0.25) (* (* r w) (* w (/ r v)))) 4.5))
     (if (<= v 3e+108)
       (-
        (+ t_1 (* (* w (* r (+ (* v -0.25) 0.375))) (* w (/ r (+ v -1.0)))))
        4.5)
       (+ t_0 (- -1.5 (* (* r w) (* (* r w) (* -0.25 (/ v (- 1.0 v)))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double tmp;
	if (v <= -4e+28) {
		tmp = t_1 + (((v * -0.25) * ((r * w) * (w * (r / v)))) - 4.5);
	} else if (v <= 3e+108) {
		tmp = (t_1 + ((w * (r * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = t_0 + (-1.5 - ((r * w) * ((r * w) * (-0.25 * (v / (1.0 - v))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = 3.0d0 + t_0
    if (v <= (-4d+28)) then
        tmp = t_1 + (((v * (-0.25d0)) * ((r * w) * (w * (r / v)))) - 4.5d0)
    else if (v <= 3d+108) then
        tmp = (t_1 + ((w * (r * ((v * (-0.25d0)) + 0.375d0))) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    else
        tmp = t_0 + ((-1.5d0) - ((r * w) * ((r * w) * ((-0.25d0) * (v / (1.0d0 - v))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double tmp;
	if (v <= -4e+28) {
		tmp = t_1 + (((v * -0.25) * ((r * w) * (w * (r / v)))) - 4.5);
	} else if (v <= 3e+108) {
		tmp = (t_1 + ((w * (r * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = t_0 + (-1.5 - ((r * w) * ((r * w) * (-0.25 * (v / (1.0 - v))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = 3.0 + t_0
	tmp = 0
	if v <= -4e+28:
		tmp = t_1 + (((v * -0.25) * ((r * w) * (w * (r / v)))) - 4.5)
	elif v <= 3e+108:
		tmp = (t_1 + ((w * (r * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5
	else:
		tmp = t_0 + (-1.5 - ((r * w) * ((r * w) * (-0.25 * (v / (1.0 - v))))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(3.0 + t_0)
	tmp = 0.0
	if (v <= -4e+28)
		tmp = Float64(t_1 + Float64(Float64(Float64(v * -0.25) * Float64(Float64(r * w) * Float64(w * Float64(r / v)))) - 4.5));
	elseif (v <= 3e+108)
		tmp = Float64(Float64(t_1 + Float64(Float64(w * Float64(r * Float64(Float64(v * -0.25) + 0.375))) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(Float64(r * w) * Float64(-0.25 * Float64(v / Float64(1.0 - v)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = 3.0 + t_0;
	tmp = 0.0;
	if (v <= -4e+28)
		tmp = t_1 + (((v * -0.25) * ((r * w) * (w * (r / v)))) - 4.5);
	elseif (v <= 3e+108)
		tmp = (t_1 + ((w * (r * ((v * -0.25) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	else
		tmp = t_0 + (-1.5 - ((r * w) * ((r * w) * (-0.25 * (v / (1.0 - v))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + t$95$0), $MachinePrecision]}, If[LessEqual[v, -4e+28], N[(t$95$1 + N[(N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 3e+108], N[(N[(t$95$1 + N[(N[(w * N[(r * 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[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(-0.25 * N[(v / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.99999999999999983e28

   1. Initial program 79.2%

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

   3. Simplified 84.1%

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

      1. associate-/l* 84.1%

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

      2. *-commutative 84.1%

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

      3. associate-*r/ 84.1%

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

      4. associate-*l* 98.1%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in v around inf 99.7%

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

       1. associate-*r/ 99.7%

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

       2. neg-mul-1 99.7%

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

   12. Simplified 99.7%

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

   if -3.99999999999999983e28 < v < 2.99999999999999984e108

   1. Initial program 85.2%

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

      1. associate-/l* 85.8%

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

      2. cancel-sign-sub-inv 85.8%

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

      3. metadata-eval 85.8%

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

      4. +-commutative 85.8%

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

      5. *-commutative 85.8%

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

      6. fma-undefine 85.8%

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

      7. *-commutative 85.8%

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

      8. *-commutative 85.8%

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

      9. associate-/l* 85.8%

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

      10. *-commutative 85.8%

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

      11. associate-*r/ 85.8%

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

      12. associate-*r* 85.8%

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

      13. associate-*l* 98.0%

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

      14. associate-*r* 99.8%

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

   4. Applied egg-rr 99.8%

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

   if 2.99999999999999984e108 < v

   1. Initial program 81.2%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in v around inf 89.0%

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

      1. *-commutative 89.0%

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

   7. Simplified 89.0%

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

   8. Taylor expanded in r around 0 76.6%

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

      1. associate-*r/ 76.6%

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

      2. *-commutative 76.6%

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

      3. associate-*r* 78.9%

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

      4. associate-*l* 78.9%

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

      5. *-commutative 78.9%

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

      6. associate-*l* 78.9%

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

      7. unpow2 78.9%

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

      8. unpow2 78.9%

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

      9. swap-sqr 86.0%

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

      10. unpow2 86.0%

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

      11. *-commutative 86.0%

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

   10. Simplified 86.0%

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

       1. associate-/l* 99.9%

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

       2. unpow2 99.9%

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

       3. associate-*l* 99.9%

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

       4. *-commutative 99.9%

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

       5. *-un-lft-identity 99.9%

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

       6. times-frac 99.9%

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

       7. metadata-eval 99.9%

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

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot \frac{v}{1 - v}\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{+28}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v}\right)\right) - 4.5\right)\\ \mathbf{elif}\;v \leq 3 \cdot 10^{+108}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(w \cdot \left(r \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot \frac{v}{1 - v}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (3.0 + (2.0 * pow(r, -2.0))) - (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (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 ** (-2.0d0)))) - (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (w * (r / (1.0d0 - v))))) + 4.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

3. Simplified 85.8%

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

   1. associate-/l* 85.6%

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

   2. *-commutative 85.6%

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

   3. associate-*r/ 85.5%

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

   4. associate-*l* 97.7%

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

   5. associate-*r* 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. div-inv 99.4%

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

   3. pow2 99.4%

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

   4. pow-flip 99.5%

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

   5. metadata-eval 99.5%

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

8. Applied egg-rr 99.5%

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

   1. *-lft-identity 99.5%

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

10. Simplified 99.5%

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

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6800000 \lor \neg \left(v \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;t\_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot \frac{v}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) + \left(w \cdot \left(r \cdot 0.375\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -6800000.0) (not (<= v 2.3e-9)))
     (+ t_0 (+ -1.5 (* (* r w) (* (* r w) (* -0.25 (/ v (+ v -1.0)))))))
     (- (+ (+ 3.0 t_0) (* (* w (* r 0.375)) (* w (/ r (+ v -1.0))))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -6800000.0) || !(v <= 2.3e-9)) {
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))));
	} else {
		tmp = ((3.0 + t_0) + ((w * (r * 0.375)) * (w * (r / (v + -1.0))))) - 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) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-6800000.0d0)) .or. (.not. (v <= 2.3d-9))) then
        tmp = t_0 + ((-1.5d0) + ((r * w) * ((r * w) * ((-0.25d0) * (v / (v + (-1.0d0)))))))
    else
        tmp = ((3.0d0 + t_0) + ((w * (r * 0.375d0)) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -6800000.0) || !(v <= 2.3e-9)) {
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))));
	} else {
		tmp = ((3.0 + t_0) + ((w * (r * 0.375)) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -6800000.0) or not (v <= 2.3e-9):
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))))
	else:
		tmp = ((3.0 + t_0) + ((w * (r * 0.375)) * (w * (r / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -6800000.0) || !(v <= 2.3e-9))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * Float64(-0.25 * Float64(v / Float64(v + -1.0)))))));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) + Float64(Float64(w * Float64(r * 0.375)) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -6800000.0) || ~((v <= 2.3e-9)))
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))));
	else
		tmp = ((3.0 + t_0) + ((w * (r * 0.375)) * (w * (r / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -6800000.0], N[Not[LessEqual[v, 2.3e-9]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(-0.25 * N[(v / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -6.8e6 or 2.2999999999999999e-9 < v

   1. Initial program 81.2%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in v around inf 85.9%

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

      1. *-commutative 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in r around 0 75.6%

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

      1. associate-*r/ 75.6%

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

      2. *-commutative 75.6%

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

      3. associate-*r* 77.2%

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

      4. associate-*l* 77.2%

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

      5. *-commutative 77.2%

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

      6. associate-*l* 77.2%

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

      7. unpow2 77.2%

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

      8. unpow2 77.2%

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

      9. swap-sqr 89.0%

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

      10. unpow2 89.0%

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

      11. *-commutative 89.0%

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

   10. Simplified 89.0%

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

       1. associate-/l* 98.9%

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

       2. unpow2 98.9%

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

       3. associate-*l* 98.9%

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

       4. *-commutative 98.9%

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

       5. *-un-lft-identity 98.9%

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

       6. times-frac 98.9%

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

       7. metadata-eval 98.9%

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

   12. Applied egg-rr 98.9%

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

   if -6.8e6 < v < 2.2999999999999999e-9

   1. Initial program 85.2%

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

      1. associate-/l* 85.2%

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

      2. cancel-sign-sub-inv 85.2%

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

      3. metadata-eval 85.2%

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

      4. +-commutative 85.2%

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

      5. *-commutative 85.2%

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

      6. fma-undefine 85.2%

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

      7. *-commutative 85.2%

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

      8. *-commutative 85.2%

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

      9. associate-/l* 85.2%

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

      10. *-commutative 85.2%

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

      11. associate-*r/ 85.2%

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

      12. associate-*r* 85.2%

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

      13. associate-*l* 97.7%

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

      14. associate-*r* 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6800000 \lor \neg \left(v \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot \frac{v}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(w \cdot \left(r \cdot 0.375\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6800000 \lor \neg \left(v \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;t\_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot \frac{v}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -6800000.0) (not (<= v 2.3e-9)))
     (+ t_0 (+ -1.5 (* (* r w) (* (* r w) (* -0.25 (/ v (+ v -1.0)))))))
     (+ t_0 (+ -1.5 (* 0.375 (* (* r w) (/ (* r w) (+ v -1.0)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -6800000.0) || !(v <= 2.3e-9)) {
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))));
	} else {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-6800000.0d0)) .or. (.not. (v <= 2.3d-9))) then
        tmp = t_0 + ((-1.5d0) + ((r * w) * ((r * w) * ((-0.25d0) * (v / (v + (-1.0d0)))))))
    else
        tmp = t_0 + ((-1.5d0) + (0.375d0 * ((r * w) * ((r * w) / (v + (-1.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -6800000.0) || !(v <= 2.3e-9)) {
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))));
	} else {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -6800000.0) or not (v <= 2.3e-9):
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))))
	else:
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -6800000.0) || !(v <= 2.3e-9))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * Float64(-0.25 * Float64(v / Float64(v + -1.0)))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.375 * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(v + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -6800000.0) || ~((v <= 2.3e-9)))
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))));
	else
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -6800000.0], N[Not[LessEqual[v, 2.3e-9]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(-0.25 * N[(v / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -6.8e6 or 2.2999999999999999e-9 < v

   1. Initial program 81.2%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in v around inf 85.9%

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

      1. *-commutative 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in r around 0 75.6%

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

      1. associate-*r/ 75.6%

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

      2. *-commutative 75.6%

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

      3. associate-*r* 77.2%

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

      4. associate-*l* 77.2%

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

      5. *-commutative 77.2%

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

      6. associate-*l* 77.2%

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

      7. unpow2 77.2%

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

      8. unpow2 77.2%

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

      9. swap-sqr 89.0%

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

      10. unpow2 89.0%

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

      11. *-commutative 89.0%

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

   10. Simplified 89.0%

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

       1. associate-/l* 98.9%

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

       2. unpow2 98.9%

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

       3. associate-*l* 98.9%

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

       4. *-commutative 98.9%

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

       5. *-un-lft-identity 98.9%

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

       6. times-frac 98.9%

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

       7. metadata-eval 98.9%

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

   12. Applied egg-rr 98.9%

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

   if -6.8e6 < v < 2.2999999999999999e-9

   1. Initial program 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in v around 0 85.2%

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

      1. add-sqr-sqrt 85.2%

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

      2. *-un-lft-identity 85.2%

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

      3. times-frac 85.2%

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

      4. associate-*r* 76.0%

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

      5. sqrt-prod 75.9%

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

      6. sqrt-prod 39.0%

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

      7. add-sqr-sqrt 66.3%

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

      8. sqrt-prod 35.7%

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

      9. add-sqr-sqrt 66.4%

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

      10. associate-*r* 55.3%

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

      11. sqrt-prod 55.3%

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

      12. sqrt-prod 31.7%

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

      13. add-sqr-sqrt 70.3%

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

      14. sqrt-prod 55.5%

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

      15. add-sqr-sqrt 99.7%

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

   7. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(0.375 \cdot \color{blue}{\left(\frac{r \cdot w}{1} \cdot \frac{r \cdot w}{v + -1}\right)} + -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6800000 \lor \neg \left(v \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot \frac{v}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.24 \cdot 10^{+67} \lor \neg \left(v \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;t\_0 + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1.24e+67) (not (<= v 2.3e-9)))
     (+ t_0 (+ -1.5 (* (* v -0.25) (/ (* r (* r (* w w))) (+ v -1.0)))))
     (+ t_0 (+ -1.5 (* 0.375 (* (* r w) (/ (* r w) (+ v -1.0)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.24e+67) || !(v <= 2.3e-9)) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-1.24d+67)) .or. (.not. (v <= 2.3d-9))) then
        tmp = t_0 + ((-1.5d0) + ((v * (-0.25d0)) * ((r * (r * (w * w))) / (v + (-1.0d0)))))
    else
        tmp = t_0 + ((-1.5d0) + (0.375d0 * ((r * w) * ((r * w) / (v + (-1.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.24e+67) || !(v <= 2.3e-9)) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1.24e+67) or not (v <= 2.3e-9):
		tmp = t_0 + (-1.5 + ((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))))
	else:
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1.24e+67) || !(v <= 2.3e-9))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(v * -0.25) * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(v + -1.0)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.375 * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(v + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -1.24e+67) || ~((v <= 2.3e-9)))
		tmp = t_0 + (-1.5 + ((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))));
	else
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1.24e+67], N[Not[LessEqual[v, 2.3e-9]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.24000000000000007e67 or 2.2999999999999999e-9 < v

   1. Initial program 81.7%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in v around inf 87.8%

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

      1. *-commutative 87.2%

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

   7. Simplified 87.8%

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

   if -1.24000000000000007e67 < v < 2.2999999999999999e-9

   1. Initial program 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in v around 0 83.2%

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

      1. add-sqr-sqrt 83.2%

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

      2. *-un-lft-identity 83.2%

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

      3. times-frac 83.2%

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

      4. associate-*r* 74.8%

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

      5. sqrt-prod 74.8%

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

      6. sqrt-prod 37.7%

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

      7. add-sqr-sqrt 64.6%

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

      8. sqrt-prod 36.1%

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

      9. add-sqr-sqrt 66.2%

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

      10. associate-*r* 56.0%

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

      11. sqrt-prod 56.0%

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

      12. sqrt-prod 31.0%

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

      13. add-sqr-sqrt 68.2%

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

      14. sqrt-prod 54.8%

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

      15. add-sqr-sqrt 97.8%

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

   7. Applied egg-rr 97.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(0.375 \cdot \color{blue}{\left(\frac{r \cdot w}{1} \cdot \frac{r \cdot w}{v + -1}\right)} + -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.24 \cdot 10^{+67} \lor \neg \left(v \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 3 + t\_0\\ \mathbf{if}\;v \leq -6800000:\\ \;\;\;\;t\_1 + \left(\left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v}\right)\right) - 4.5\right)\\ \mathbf{elif}\;v \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\left(t\_1 + \left(w \cdot \left(r \cdot 0.375\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot \frac{v}{v + -1}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ 3.0 t_0)))
   (if (<= v -6800000.0)
     (+ t_1 (- (* (* v -0.25) (* (* r w) (* w (/ r v)))) 4.5))
     (if (<= v 2.3e-9)
       (- (+ t_1 (* (* w (* r 0.375)) (* w (/ r (+ v -1.0))))) 4.5)
       (+ t_0 (+ -1.5 (* (* r w) (* (* r w) (* -0.25 (/ v (+ v -1.0)))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double tmp;
	if (v <= -6800000.0) {
		tmp = t_1 + (((v * -0.25) * ((r * w) * (w * (r / v)))) - 4.5);
	} else if (v <= 2.3e-9) {
		tmp = (t_1 + ((w * (r * 0.375)) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = 3.0d0 + t_0
    if (v <= (-6800000.0d0)) then
        tmp = t_1 + (((v * (-0.25d0)) * ((r * w) * (w * (r / v)))) - 4.5d0)
    else if (v <= 2.3d-9) then
        tmp = (t_1 + ((w * (r * 0.375d0)) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    else
        tmp = t_0 + ((-1.5d0) + ((r * w) * ((r * w) * ((-0.25d0) * (v / (v + (-1.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double tmp;
	if (v <= -6800000.0) {
		tmp = t_1 + (((v * -0.25) * ((r * w) * (w * (r / v)))) - 4.5);
	} else if (v <= 2.3e-9) {
		tmp = (t_1 + ((w * (r * 0.375)) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = 3.0 + t_0
	tmp = 0
	if v <= -6800000.0:
		tmp = t_1 + (((v * -0.25) * ((r * w) * (w * (r / v)))) - 4.5)
	elif v <= 2.3e-9:
		tmp = (t_1 + ((w * (r * 0.375)) * (w * (r / (v + -1.0))))) - 4.5
	else:
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(3.0 + t_0)
	tmp = 0.0
	if (v <= -6800000.0)
		tmp = Float64(t_1 + Float64(Float64(Float64(v * -0.25) * Float64(Float64(r * w) * Float64(w * Float64(r / v)))) - 4.5));
	elseif (v <= 2.3e-9)
		tmp = Float64(Float64(t_1 + Float64(Float64(w * Float64(r * 0.375)) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * Float64(-0.25 * Float64(v / Float64(v + -1.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = 3.0 + t_0;
	tmp = 0.0;
	if (v <= -6800000.0)
		tmp = t_1 + (((v * -0.25) * ((r * w) * (w * (r / v)))) - 4.5);
	elseif (v <= 2.3e-9)
		tmp = (t_1 + ((w * (r * 0.375)) * (w * (r / (v + -1.0))))) - 4.5;
	else
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * (-0.25 * (v / (v + -1.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + t$95$0), $MachinePrecision]}, If[LessEqual[v, -6800000.0], N[(t$95$1 + N[(N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2.3e-9], N[(N[(t$95$1 + N[(N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(-0.25 * N[(v / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -6.8e6

   1. Initial program 80.5%

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

   3. Simplified 85.1%

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

      1. associate-/l* 85.1%

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

      2. *-commutative 85.1%

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

      3. associate-*r/ 85.1%

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

      4. associate-*l* 98.1%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 99.1%

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

      1. *-commutative 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in v around inf 99.2%

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

       1. associate-*r/ 99.2%

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

       2. neg-mul-1 99.2%

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

   12. Simplified 99.2%

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

   if -6.8e6 < v < 2.2999999999999999e-9

   1. Initial program 85.2%

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

      1. associate-/l* 85.2%

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

      2. cancel-sign-sub-inv 85.2%

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

      3. metadata-eval 85.2%

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

      4. +-commutative 85.2%

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

      5. *-commutative 85.2%

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

      6. fma-undefine 85.2%

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

      7. *-commutative 85.2%

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

      8. *-commutative 85.2%

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

      9. associate-/l* 85.2%

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

      10. *-commutative 85.2%

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

      11. associate-*r/ 85.2%

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

      12. associate-*r* 85.2%

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

      13. associate-*l* 97.7%

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

      14. associate-*r* 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   if 2.2999999999999999e-9 < v

   1. Initial program 81.9%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in v around inf 87.2%

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

      1. *-commutative 87.2%

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

   7. Simplified 87.2%

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

   8. Taylor expanded in r around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*r* 78.0%

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

      4. associate-*l* 78.0%

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

      5. *-commutative 78.0%

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

      6. associate-*l* 78.0%

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

      7. unpow2 78.0%

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

      8. unpow2 78.0%

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

      9. swap-sqr 88.5%

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

      10. unpow2 88.5%

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

      11. *-commutative 88.5%

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

   10. Simplified 88.5%

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

       1. associate-/l* 98.8%

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

       2. unpow2 98.8%

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

       3. associate-*l* 98.8%

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

       4. *-commutative 98.8%

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

       5. *-un-lft-identity 98.8%

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

       6. times-frac 98.8%

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

       7. metadata-eval 98.8%

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

   12. Applied egg-rr 98.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot \frac{v}{1 - v}\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6800000:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v}\right)\right) - 4.5\right)\\ \mathbf{elif}\;v \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(w \cdot \left(r \cdot 0.375\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot \frac{v}{v + -1}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

Python

def code(v, w, r):
	return (3.0 + (2.0 / (r * r))) + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (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(-2.0 * v))) * Float64(Float64(r * w) * Float64(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 + (-2.0 * v))) * ((r * w) * (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[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.3%

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

3. Simplified 85.8%

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

   1. associate-/l* 85.6%

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

   2. *-commutative 85.6%

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

   3. associate-*r/ 85.5%

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

   4. associate-*l* 97.7%

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

   5. associate-*r* 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 8: 87.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.35e-63)
   (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (* (* r w) (/ (* r w) (+ v -1.0))))))
   (+ (* (/ 2.0 r) (/ 1.0 r)) (- -1.5 (* 0.375 (* r (* r (* w w))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.35e-63) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))));
	} else {
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * (r * (r * (w * w)))));
	}
	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 (r <= 2.35d-63) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * ((r * w) * ((r * w) / (v + (-1.0d0))))))
    else
        tmp = ((2.0d0 / r) * (1.0d0 / r)) + ((-1.5d0) - (0.375d0 * (r * (r * (w * w)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.35e-63) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))));
	} else {
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * (r * (r * (w * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.35e-63:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))))
	else:
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * (r * (r * (w * w)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.35e-63)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(v + -1.0))))));
	else
		tmp = Float64(Float64(Float64(2.0 / r) * Float64(1.0 / r)) + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.35e-63)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * w) * ((r * w) / (v + -1.0)))));
	else
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * (r * (r * (w * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.35e-63], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.35e-63

   1. Initial program 80.9%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in v around 0 77.6%

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

      1. add-sqr-sqrt 77.6%

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

      2. *-un-lft-identity 77.6%

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

      3. times-frac 77.6%

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

      4. associate-*r* 70.5%

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

      5. sqrt-prod 70.5%

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

      6. sqrt-prod 20.0%

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

      7. add-sqr-sqrt 64.0%

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

      8. sqrt-prod 36.5%

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

      9. add-sqr-sqrt 73.2%

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

      10. associate-*r* 63.3%

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

      11. sqrt-prod 63.3%

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

      12. sqrt-prod 20.7%

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

      13. add-sqr-sqrt 69.1%

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

      14. sqrt-prod 48.0%

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

      15. add-sqr-sqrt 89.3%

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

   7. Applied egg-rr 89.3%

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

   if 2.35e-63 < r

   1. Initial program 88.4%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in v around 0 68.1%

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

      1. associate-/r* 68.1%

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

      2. div-inv 68.1%

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

   7. Applied egg-rr 68.1%

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

   8. Taylor expanded in v around 0 85.4%

      \[\leadsto \frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{r}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.25 \cdot 10^{-95}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.25e-95)
   (- (+ 3.0 (/ 2.0 (* r r))) (+ 4.5 (* (* v -0.25) (* (* r w) (* r w)))))
   (+ (* (/ 2.0 r) (/ 1.0 r)) (- -1.5 (* 0.375 (* r (* r (* w w))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.25e-95) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + ((v * -0.25) * ((r * w) * (r * w))));
	} else {
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * (r * (r * (w * w)))));
	}
	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 (r <= 1.25d-95) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - (4.5d0 + ((v * (-0.25d0)) * ((r * w) * (r * w))))
    else
        tmp = ((2.0d0 / r) * (1.0d0 / r)) + ((-1.5d0) - (0.375d0 * (r * (r * (w * w)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.25e-95) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + ((v * -0.25) * ((r * w) * (r * w))));
	} else {
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * (r * (r * (w * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.25e-95:
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + ((v * -0.25) * ((r * w) * (r * w))))
	else:
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * (r * (r * (w * w)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.25e-95)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(4.5 + Float64(Float64(v * -0.25) * Float64(Float64(r * w) * Float64(r * w)))));
	else
		tmp = Float64(Float64(Float64(2.0 / r) * Float64(1.0 / r)) + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.25e-95)
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + ((v * -0.25) * ((r * w) * (r * w))));
	else
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * (r * (r * (w * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.25e-95], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.5 + N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.2499999999999999e-95

   1. Initial program 81.0%

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

   3. Simplified 82.1%

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

      1. associate-/l* 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*r/ 82.1%

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

      4. associate-*l* 97.6%

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

      5. associate-*r* 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in v around inf 87.6%

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

      1. *-commutative 87.6%

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

   9. Simplified 87.6%

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

   10. Taylor expanded in v around 0 74.5%

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

   if 1.2499999999999999e-95 < r

   1. Initial program 87.7%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in v around 0 69.6%

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

      1. associate-/r* 69.6%

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

      2. div-inv 69.6%

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

   7. Applied egg-rr 69.6%

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

   8. Taylor expanded in v around 0 86.0%

      \[\leadsto \frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{r}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.25 \cdot 10^{-95}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(r \cdot \left(w \cdot w\right)\right)\\ \mathbf{if}\;r \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \frac{t\_0}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - 0.375 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* r (* w w)))))
   (if (<= r 2e-64)
     (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (/ t_0 (+ v -1.0)))))
     (+ (* (/ 2.0 r) (/ 1.0 r)) (- -1.5 (* 0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double tmp;
	if (r <= 2e-64) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (t_0 / (v + -1.0))));
	} else {
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * (r * (w * w))
    if (r <= 2d-64) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * (t_0 / (v + (-1.0d0)))))
    else
        tmp = ((2.0d0 / r) * (1.0d0 / r)) + ((-1.5d0) - (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double tmp;
	if (r <= 2e-64) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (t_0 / (v + -1.0))));
	} else {
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * (r * (w * w))
	tmp = 0
	if r <= 2e-64:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (t_0 / (v + -1.0))))
	else:
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(r * Float64(w * w)))
	tmp = 0.0
	if (r <= 2e-64)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(t_0 / Float64(v + -1.0)))));
	else
		tmp = Float64(Float64(Float64(2.0 / r) * Float64(1.0 / r)) + Float64(-1.5 - Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * (r * (w * w));
	tmp = 0.0;
	if (r <= 2e-64)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (t_0 / (v + -1.0))));
	else
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2e-64], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(t$95$0 / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.99999999999999993e-64

   1. Initial program 80.9%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in v around 0 77.6%

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

   if 1.99999999999999993e-64 < r

   1. Initial program 88.4%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in v around 0 68.1%

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

      1. associate-/r* 68.1%

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

      2. div-inv 68.1%

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

   7. Applied egg-rr 68.1%

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

   8. Taylor expanded in v around 0 85.4%

      \[\leadsto \frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{r}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / r) * (1.0d0 / r)) + ((-1.5d0) - (0.375d0 * (r * (r * (w * w)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

3. Simplified 85.5%

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

5. Taylor expanded in v around 0 73.0%

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

   1. associate-/r* 73.0%

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

   2. div-inv 73.0%

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

7. Applied egg-rr 73.0%

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

8. Taylor expanded in v around 0 81.6%

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

9. Final simplification 81.6%

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

Reproduce

herbie shell --seed 2024106 
(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))