Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 98.6%

Time: 14.2s

Alternatives: 13

Speedup: 1.7×

• 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: 84.5% 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: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2 \cdot 10^{+127} \lor \neg \left(v \leq 5 \cdot 10^{-56}\right):\\ \;\;\;\;t\_0 + \left(-1.5 + 0.25 \cdot \left(\frac{r}{\frac{1}{w}} \cdot \frac{w}{\frac{-1}{r}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + t\_0\right) - \left(4.5 + \frac{r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{\frac{1 - v}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -2e+127) (not (<= v 5e-56)))
     (+ t_0 (+ -1.5 (* 0.25 (* (/ r (/ 1.0 w)) (/ w (/ -1.0 r))))))
     (-
      (+ 3.0 t_0)
      (+ 4.5 (/ (* r (* w (+ 0.375 (* v -0.25)))) (/ (- 1.0 v) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2e+127) || !(v <= 5e-56)) {
		tmp = t_0 + (-1.5 + (0.25 * ((r / (1.0 / w)) * (w / (-1.0 / r)))));
	} else {
		tmp = (3.0 + t_0) - (4.5 + ((r * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / (r * 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-2d+127)) .or. (.not. (v <= 5d-56))) then
        tmp = t_0 + ((-1.5d0) + (0.25d0 * ((r / (1.0d0 / w)) * (w / ((-1.0d0) / r)))))
    else
        tmp = (3.0d0 + t_0) - (4.5d0 + ((r * (w * (0.375d0 + (v * (-0.25d0))))) / ((1.0d0 - v) / (r * w))))
    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 <= -2e+127) || !(v <= 5e-56)) {
		tmp = t_0 + (-1.5 + (0.25 * ((r / (1.0 / w)) * (w / (-1.0 / r)))));
	} else {
		tmp = (3.0 + t_0) - (4.5 + ((r * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2e+127) or not (v <= 5e-56):
		tmp = t_0 + (-1.5 + (0.25 * ((r / (1.0 / w)) * (w / (-1.0 / r)))))
	else:
		tmp = (3.0 + t_0) - (4.5 + ((r * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2e+127) || !(v <= 5e-56))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.25 * Float64(Float64(r / Float64(1.0 / w)) * Float64(w / Float64(-1.0 / r))))));
	else
		tmp = Float64(Float64(3.0 + t_0) - Float64(4.5 + Float64(Float64(r * Float64(w * Float64(0.375 + Float64(v * -0.25)))) / Float64(Float64(1.0 - v) / Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2e+127) || ~((v <= 5e-56)))
		tmp = t_0 + (-1.5 + (0.25 * ((r / (1.0 / w)) * (w / (-1.0 / r)))));
	else
		tmp = (3.0 + t_0) - (4.5 + ((r * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / (r * w))));
	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, -2e+127], N[Not[LessEqual[v, 5e-56]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(0.25 * N[(N[(r / N[(1.0 / w), $MachinePrecision]), $MachinePrecision] * N[(w / N[(-1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(4.5 + N[(N[(r * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.99999999999999991e127 or 4.99999999999999997e-56 < v

   1. Initial program 85.5%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in v around inf 81.3%

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

      1. *-commutative 81.3%

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

      2. *-commutative 81.3%

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

      3. unpow2 81.3%

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

      4. unpow2 81.3%

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

      5. swap-sqr 99.9%

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

      6. unpow2 99.9%

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

      7. *-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. pow2 99.9%

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

      3. remove-double-div 99.9%

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

      4. un-div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. inv-pow 99.8%

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

       2. unpow-prod-down 99.9%

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

       3. inv-pow 99.9%

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

       4. inv-pow 99.9%

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

   11. Applied egg-rr 99.9%

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

       1. *-commutative 99.9%

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

       2. times-frac 99.9%

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

   13. Applied egg-rr 99.9%

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

   if -1.99999999999999991e127 < v < 4.99999999999999997e-56

   1. Initial program 89.0%

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

      1. associate--l- 89.0%

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

      2. associate-*l* 84.9%

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

      3. sqr-neg 84.9%

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

      4. associate-*l* 89.0%

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

      5. associate-/l* 89.6%

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

      6. fma-define 89.6%

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

   3. Simplified 89.6%

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

      1. add-sqr-sqrt 89.6%

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

      2. *-un-lft-identity 89.6%

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

      3. times-frac 89.6%

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

      4. *-commutative 89.6%

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

      5. sqrt-prod 46.5%

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

      6. *-commutative 46.5%

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

      7. sqrt-prod 46.5%

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

      8. sqrt-prod 25.0%

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

      9. add-sqr-sqrt 36.6%

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

      10. associate-*r* 36.6%

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

      11. add-sqr-sqrt 69.1%

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

   6. Applied egg-rr 99.8%

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

      1. /-rgt-identity 99.8%

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

      2. associate-*r* 99.8%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\color{blue}{0.375} + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      7. associate-*r* 99.8%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.375 + \color{blue}{\left(0.125 \cdot -2\right) \cdot v}\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      8. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in w around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+127} \lor \neg \left(v \leq 5 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.25 \cdot \left(\frac{r}{\frac{1}{w}} \cdot \frac{w}{\frac{-1}{r}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \frac{r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{\frac{1 - v}{r \cdot w}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2000000000000 \lor \neg \left(v \leq 5 \cdot 10^{-56}\right):\\ \;\;\;\;t\_0 + \left(-1.5 - \left(\frac{w}{\frac{1}{r}} \cdot \frac{r}{\frac{1}{w}}\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.375 \cdot \frac{r \cdot w}{\frac{1}{w} \cdot \frac{-1}{r}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -2000000000000.0) (not (<= v 5e-56)))
     (+ t_0 (- -1.5 (* (* (/ w (/ 1.0 r)) (/ r (/ 1.0 w))) 0.25)))
     (+ t_0 (+ -1.5 (* 0.375 (/ (* r w) (* (/ 1.0 w) (/ -1.0 r)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2000000000000.0) || !(v <= 5e-56)) {
		tmp = t_0 + (-1.5 - (((w / (1.0 / r)) * (r / (1.0 / w))) * 0.25));
	} else {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / ((1.0 / w) * (-1.0 / r)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-2000000000000.0d0)) .or. (.not. (v <= 5d-56))) then
        tmp = t_0 + ((-1.5d0) - (((w / (1.0d0 / r)) * (r / (1.0d0 / w))) * 0.25d0))
    else
        tmp = t_0 + ((-1.5d0) + (0.375d0 * ((r * w) / ((1.0d0 / w) * ((-1.0d0) / r)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2000000000000.0) || !(v <= 5e-56)) {
		tmp = t_0 + (-1.5 - (((w / (1.0 / r)) * (r / (1.0 / w))) * 0.25));
	} else {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / ((1.0 / w) * (-1.0 / r)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2000000000000.0) or not (v <= 5e-56):
		tmp = t_0 + (-1.5 - (((w / (1.0 / r)) * (r / (1.0 / w))) * 0.25))
	else:
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / ((1.0 / w) * (-1.0 / r)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2000000000000.0) || !(v <= 5e-56))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(w / Float64(1.0 / r)) * Float64(r / Float64(1.0 / w))) * 0.25)));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.375 * Float64(Float64(r * w) / Float64(Float64(1.0 / w) * Float64(-1.0 / r))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2000000000000.0) || ~((v <= 5e-56)))
		tmp = t_0 + (-1.5 - (((w / (1.0 / r)) * (r / (1.0 / w))) * 0.25));
	else
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / ((1.0 / w) * (-1.0 / r)))));
	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, -2000000000000.0], N[Not[LessEqual[v, 5e-56]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 - N[(N[(N[(w / N[(1.0 / r), $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] / N[(N[(1.0 / w), $MachinePrecision] * N[(-1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2e12 or 4.99999999999999997e-56 < v

   1. Initial program 87.0%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in v around inf 83.5%

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

      1. *-commutative 83.5%

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

      2. *-commutative 83.5%

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

      3. unpow2 83.5%

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

      4. unpow2 83.5%

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

      5. swap-sqr 99.7%

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

      6. unpow2 99.7%

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

      7. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. pow2 99.7%

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

      3. remove-double-div 99.7%

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

      4. un-div-inv 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. inv-pow 99.7%

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

       2. unpow-prod-down 99.7%

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

       3. inv-pow 99.7%

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

       4. inv-pow 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. *-commutative 99.7%

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

       2. times-frac 99.8%

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

   13. Applied egg-rr 99.8%

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

   if -2e12 < v < 4.99999999999999997e-56

   1. Initial program 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in v around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. swap-sqr 99.4%

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

      5. unpow2 99.4%

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

      6. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. *-commutative 88.3%

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

      2. pow2 88.3%

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

      3. remove-double-div 88.3%

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

      4. un-div-inv 88.3%

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

   9. Applied egg-rr 99.4%

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

       1. inv-pow 88.3%

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

       2. unpow-prod-down 88.3%

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

       3. inv-pow 88.3%

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

       4. inv-pow 88.3%

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

   11. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2000000000000 \lor \neg \left(v \leq 5 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\frac{w}{\frac{1}{r}} \cdot \frac{r}{\frac{1}{w}}\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \frac{r \cdot w}{\frac{1}{w} \cdot \frac{-1}{r}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -880000000000:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.25 \cdot \frac{r \cdot w}{\frac{\frac{-1}{w}}{r}}\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.375 \cdot \frac{r \cdot w}{\frac{1}{w} \cdot \frac{-1}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -880000000000.0)
     (+ t_0 (+ -1.5 (* 0.25 (/ (* r w) (/ (/ -1.0 w) r)))))
     (if (<= v 2.5e-55)
       (+ t_0 (+ -1.5 (* 0.375 (/ (* r w) (* (/ 1.0 w) (/ -1.0 r))))))
       (+ t_0 (- -1.5 (* 0.25 (* (* r w) (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -880000000000.0) {
		tmp = t_0 + (-1.5 + (0.25 * ((r * w) / ((-1.0 / w) / r))));
	} else if (v <= 2.5e-55) {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / ((1.0 / w) * (-1.0 / r)))));
	} else {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (v <= (-880000000000.0d0)) then
        tmp = t_0 + ((-1.5d0) + (0.25d0 * ((r * w) / (((-1.0d0) / w) / r))))
    else if (v <= 2.5d-55) then
        tmp = t_0 + ((-1.5d0) + (0.375d0 * ((r * w) / ((1.0d0 / w) * ((-1.0d0) / r)))))
    else
        tmp = t_0 + ((-1.5d0) - (0.25d0 * ((r * w) * (r * w))))
    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 <= -880000000000.0) {
		tmp = t_0 + (-1.5 + (0.25 * ((r * w) / ((-1.0 / w) / r))));
	} else if (v <= 2.5e-55) {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / ((1.0 / w) * (-1.0 / r)))));
	} else {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -880000000000.0:
		tmp = t_0 + (-1.5 + (0.25 * ((r * w) / ((-1.0 / w) / r))))
	elif v <= 2.5e-55:
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / ((1.0 / w) * (-1.0 / r)))))
	else:
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -880000000000.0)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.25 * Float64(Float64(r * w) / Float64(Float64(-1.0 / w) / r)))));
	elseif (v <= 2.5e-55)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.375 * Float64(Float64(r * w) / Float64(Float64(1.0 / w) * Float64(-1.0 / r))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -880000000000.0)
		tmp = t_0 + (-1.5 + (0.25 * ((r * w) / ((-1.0 / w) / r))));
	elseif (v <= 2.5e-55)
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / ((1.0 / w) * (-1.0 / r)))));
	else
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -880000000000.0], N[(t$95$0 + N[(-1.5 + N[(0.25 * N[(N[(r * w), $MachinePrecision] / N[(N[(-1.0 / w), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2.5e-55], N[(t$95$0 + N[(-1.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] / N[(N[(1.0 / w), $MachinePrecision] * N[(-1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -8.8e11

   1. Initial program 87.3%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in v around inf 82.0%

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

      1. *-commutative 82.0%

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

      2. *-commutative 82.0%

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

      3. unpow2 82.0%

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

      4. unpow2 82.0%

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

      5. swap-sqr 99.6%

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

      6. unpow2 99.6%

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

      7. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. *-commutative 89.2%

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

      2. pow2 89.2%

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

   9. Applied egg-rr 99.6%

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

       1. pow2 99.6%

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

       2. metadata-eval 99.6%

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

       3. pow-div 99.6%

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

       4. pow1 99.6%

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

       5. inv-pow 99.6%

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

       6. associate-/r* 99.7%

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

   11. Applied egg-rr 99.7%

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

   if -8.8e11 < v < 2.5000000000000001e-55

   1. Initial program 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in v around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. swap-sqr 99.4%

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

      5. unpow2 99.4%

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

      6. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. *-commutative 88.3%

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

      2. pow2 88.3%

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

      3. remove-double-div 88.3%

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

      4. un-div-inv 88.3%

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

   9. Applied egg-rr 99.4%

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

       1. inv-pow 88.3%

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

       2. unpow-prod-down 88.3%

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

       3. inv-pow 88.3%

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

       4. inv-pow 88.3%

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

   11. Applied egg-rr 99.4%

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

   if 2.5000000000000001e-55 < v

   1. Initial program 86.7%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in v around inf 85.0%

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

      1. *-commutative 85.0%

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

      2. *-commutative 85.0%

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

      3. unpow2 85.0%

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

      4. unpow2 85.0%

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

      5. swap-sqr 99.9%

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

      6. unpow2 99.9%

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

      7. *-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. *-commutative 91.3%

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

      2. pow2 91.3%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -880000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.25 \cdot \frac{r \cdot w}{\frac{\frac{-1}{w}}{r}}\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \frac{r \cdot w}{\frac{1}{w} \cdot \frac{-1}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2000000000000 \lor \neg \left(v \leq 2.5 \cdot 10^{-55}\right):\\ \;\;\;\;t\_0 + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.375 \cdot \frac{r \cdot w}{\frac{-1}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -2000000000000.0) (not (<= v 2.5e-55)))
     (+ t_0 (- -1.5 (* 0.25 (* (* r w) (* r w)))))
     (+ t_0 (+ -1.5 (* 0.375 (/ (* r w) (/ -1.0 (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2000000000000.0) || !(v <= 2.5e-55)) {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	} else {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / (-1.0 / (r * 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-2000000000000.0d0)) .or. (.not. (v <= 2.5d-55))) then
        tmp = t_0 + ((-1.5d0) - (0.25d0 * ((r * w) * (r * w))))
    else
        tmp = t_0 + ((-1.5d0) + (0.375d0 * ((r * w) / ((-1.0d0) / (r * w)))))
    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 <= -2000000000000.0) || !(v <= 2.5e-55)) {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	} else {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / (-1.0 / (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2000000000000.0) or not (v <= 2.5e-55):
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))))
	else:
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / (-1.0 / (r * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2000000000000.0) || !(v <= 2.5e-55))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * Float64(Float64(r * w) * Float64(r * w)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.375 * Float64(Float64(r * w) / Float64(-1.0 / Float64(r * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2000000000000.0) || ~((v <= 2.5e-55)))
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	else
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / (-1.0 / (r * w)))));
	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, -2000000000000.0], N[Not[LessEqual[v, 2.5e-55]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] / N[(-1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2e12 or 2.5000000000000001e-55 < v

   1. Initial program 87.0%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in v around inf 83.5%

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

      1. *-commutative 83.5%

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

      2. *-commutative 83.5%

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

      3. unpow2 83.5%

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

      4. unpow2 83.5%

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

      5. swap-sqr 99.7%

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

      6. unpow2 99.7%

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

      7. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. *-commutative 90.2%

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

      2. pow2 90.2%

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

   9. Applied egg-rr 99.7%

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

   if -2e12 < v < 2.5000000000000001e-55

   1. Initial program 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in v around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. swap-sqr 99.4%

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

      5. unpow2 99.4%

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

      6. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. *-commutative 88.3%

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

      2. pow2 88.3%

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

      3. remove-double-div 88.3%

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

      4. un-div-inv 88.3%

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

   9. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2000000000000 \lor \neg \left(v \leq 2.5 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \frac{r \cdot w}{\frac{-1}{r \cdot w}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -880000000000:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.25 \cdot \frac{r \cdot w}{\frac{\frac{-1}{w}}{r}}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-58}:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.375 \cdot \frac{r \cdot w}{\frac{-1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -880000000000.0)
     (+ t_0 (+ -1.5 (* 0.25 (/ (* r w) (/ (/ -1.0 w) r)))))
     (if (<= v 2e-58)
       (+ t_0 (+ -1.5 (* 0.375 (/ (* r w) (/ -1.0 (* r w))))))
       (+ t_0 (- -1.5 (* 0.25 (* (* r w) (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -880000000000.0) {
		tmp = t_0 + (-1.5 + (0.25 * ((r * w) / ((-1.0 / w) / r))));
	} else if (v <= 2e-58) {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / (-1.0 / (r * w)))));
	} else {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (v <= (-880000000000.0d0)) then
        tmp = t_0 + ((-1.5d0) + (0.25d0 * ((r * w) / (((-1.0d0) / w) / r))))
    else if (v <= 2d-58) then
        tmp = t_0 + ((-1.5d0) + (0.375d0 * ((r * w) / ((-1.0d0) / (r * w)))))
    else
        tmp = t_0 + ((-1.5d0) - (0.25d0 * ((r * w) * (r * w))))
    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 <= -880000000000.0) {
		tmp = t_0 + (-1.5 + (0.25 * ((r * w) / ((-1.0 / w) / r))));
	} else if (v <= 2e-58) {
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / (-1.0 / (r * w)))));
	} else {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -880000000000.0:
		tmp = t_0 + (-1.5 + (0.25 * ((r * w) / ((-1.0 / w) / r))))
	elif v <= 2e-58:
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / (-1.0 / (r * w)))))
	else:
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -880000000000.0)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.25 * Float64(Float64(r * w) / Float64(Float64(-1.0 / w) / r)))));
	elseif (v <= 2e-58)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.375 * Float64(Float64(r * w) / Float64(-1.0 / Float64(r * w))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -880000000000.0)
		tmp = t_0 + (-1.5 + (0.25 * ((r * w) / ((-1.0 / w) / r))));
	elseif (v <= 2e-58)
		tmp = t_0 + (-1.5 + (0.375 * ((r * w) / (-1.0 / (r * w)))));
	else
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -880000000000.0], N[(t$95$0 + N[(-1.5 + N[(0.25 * N[(N[(r * w), $MachinePrecision] / N[(N[(-1.0 / w), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2e-58], N[(t$95$0 + N[(-1.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] / N[(-1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -8.8e11

   1. Initial program 87.3%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in v around inf 82.0%

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

      1. *-commutative 82.0%

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

      2. *-commutative 82.0%

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

      3. unpow2 82.0%

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

      4. unpow2 82.0%

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

      5. swap-sqr 99.6%

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

      6. unpow2 99.6%

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

      7. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. *-commutative 89.2%

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

      2. pow2 89.2%

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

   9. Applied egg-rr 99.6%

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

       1. pow2 99.6%

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

       2. metadata-eval 99.6%

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

       3. pow-div 99.6%

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

       4. pow1 99.6%

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

       5. inv-pow 99.6%

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

       6. associate-/r* 99.7%

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

   11. Applied egg-rr 99.7%

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

   if -8.8e11 < v < 2.0000000000000001e-58

   1. Initial program 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in v around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. swap-sqr 99.4%

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

      5. unpow2 99.4%

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

      6. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. *-commutative 88.3%

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

      2. pow2 88.3%

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

      3. remove-double-div 88.3%

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

      4. un-div-inv 88.3%

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

   9. Applied egg-rr 99.4%

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

   if 2.0000000000000001e-58 < v

   1. Initial program 86.7%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in v around inf 85.0%

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

      1. *-commutative 85.0%

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

      2. *-commutative 85.0%

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

      3. unpow2 85.0%

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

      4. unpow2 85.0%

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

      5. swap-sqr 99.9%

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

      6. unpow2 99.9%

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

      7. *-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. *-commutative 91.3%

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

      2. pow2 91.3%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -880000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.25 \cdot \frac{r \cdot w}{\frac{\frac{-1}{w}}{r}}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \frac{r \cdot w}{\frac{-1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+ 3.0 (/ 2.0 (* r r)))
  (+ (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (/ (* r w) (- 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))) * ((r * w) * ((r * w) / (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))) * ((r * w) * ((r * w) / (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))) * ((r * w) * ((r * w) / (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))) * ((r * w) * ((r * w) / (1.0 - v)))) + 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(Float64(r * w) / 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))) * ((r * w) * ((r * w) / (1.0 - v)))) + 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[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.6%

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

   1. associate--l- 87.6%

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

   2. associate-*l* 82.7%

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

   3. sqr-neg 82.7%

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

   4. associate-*l* 87.6%

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

   5. associate-/l* 89.3%

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

   6. fma-define 89.3%

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

3. Simplified 89.3%

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

   1. add-sqr-sqrt 89.2%

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

   2. *-un-lft-identity 89.2%

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

   3. times-frac 89.2%

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

   4. *-commutative 89.2%

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

   5. sqrt-prod 48.2%

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

   6. *-commutative 48.2%

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

   7. sqrt-prod 48.2%

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

   8. sqrt-prod 25.3%

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

   9. add-sqr-sqrt 38.0%

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

   10. associate-*r* 38.0%

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

   11. add-sqr-sqrt 72.2%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 7: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

   1. associate--l- 87.6%

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

   2. associate-*l* 82.7%

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

   3. sqr-neg 82.7%

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

   4. associate-*l* 87.6%

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

   5. associate-/l* 89.3%

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

   6. fma-define 89.3%

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

3. Simplified 89.3%

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

   1. div-inv 89.3%

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

   2. *-commutative 89.3%

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

   3. associate-*r* 89.2%

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

   4. div-inv 89.2%

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

   5. *-commutative 89.2%

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

   6. associate-*l* 97.6%

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

   7. add-sqr-sqrt 52.3%

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

   8. associate-*r* 52.3%

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

   9. add-sqr-sqrt 28.1%

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

   10. sqrt-prod 38.0%

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

   11. sqrt-prod 38.0%

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

   12. *-commutative 38.0%

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

   13. sqrt-prod 72.2%

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

   14. *-commutative 72.2%

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

   15. associate-*l* 72.2%

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

6. Applied egg-rr 98.0%

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

7. Final simplification 98.0%

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

Alternative 8: 99.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (+ 3.0 (/ 2.0 (* r r)))
  (- (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (/ w (/ (+ v -1.0) r)))) 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 / ((v + -1.0) / r)))) - 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 / ((v + (-1.0d0)) / r)))) - 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 / ((v + -1.0) / r)))) - 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 / ((v + -1.0) / r)))) - 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(Float64(v + -1.0) / r)))) - 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 / ((v + -1.0) / r)))) - 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[(N[(v + -1.0), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.6%

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

   1. associate--l- 87.6%

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

   2. associate-*l* 82.7%

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

   3. sqr-neg 82.7%

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

   4. associate-*l* 87.6%

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

   5. associate-/l* 89.3%

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

   6. fma-define 89.3%

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

3. Simplified 89.3%

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

   1. associate-/l* 89.3%

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

   2. *-commutative 89.3%

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

   3. associate-*r/ 89.2%

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

   4. associate-*l* 97.8%

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

   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. add-sqr-sqrt 53.0%

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

   7. associate-*l* 53.0%

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

   8. add-sqr-sqrt 28.0%

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

   9. sqrt-prod 38.0%

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

   10. sqrt-prod 38.0%

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

   11. sqrt-prod 72.2%

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

   12. *-commutative 72.2%

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

   13. sqrt-prod 38.0%

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

   14. *-commutative 38.0%

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

   15. sqrt-prod 38.0%

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

   16. sqrt-prod 28.0%

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

   17. add-sqr-sqrt 53.0%

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

   18. associate-*r* 53.0%

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

   19. add-sqr-sqrt 99.4%

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

   20. clear-num 99.4%

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

   21. un-div-inv 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 9: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -880000000000 \lor \neg \left(v \leq 2 \cdot 10^{-58}\right):\\ \;\;\;\;t\_1 + \left(-1.5 - 0.25 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 - 0.375 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -880000000000.0) (not (<= v 2e-58)))
     (+ t_1 (- -1.5 (* 0.25 t_0)))
     (+ t_1 (- -1.5 (* 0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -880000000000.0) || !(v <= 2e-58)) {
		tmp = t_1 + (-1.5 - (0.25 * t_0));
	} else {
		tmp = t_1 + (-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) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-880000000000.0d0)) .or. (.not. (v <= 2d-58))) then
        tmp = t_1 + ((-1.5d0) - (0.25d0 * t_0))
    else
        tmp = t_1 + ((-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 * w) * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -880000000000.0) || !(v <= 2e-58)) {
		tmp = t_1 + (-1.5 - (0.25 * t_0));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -880000000000.0) or not (v <= 2e-58):
		tmp = t_1 + (-1.5 - (0.25 * t_0))
	else:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -880000000000.0) || !(v <= 2e-58))
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.25 * t_0)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -880000000000.0) || ~((v <= 2e-58)))
		tmp = t_1 + (-1.5 - (0.25 * t_0));
	else
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -880000000000.0], N[Not[LessEqual[v, 2e-58]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 - N[(0.25 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -8.8e11 or 2.0000000000000001e-58 < v

   1. Initial program 87.0%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in v around inf 83.5%

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

      1. *-commutative 83.5%

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

      2. *-commutative 83.5%

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

      3. unpow2 83.5%

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

      4. unpow2 83.5%

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

      5. swap-sqr 99.7%

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

      6. unpow2 99.7%

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

      7. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. *-commutative 90.2%

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

      2. pow2 90.2%

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

   9. Applied egg-rr 99.7%

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

   if -8.8e11 < v < 2.0000000000000001e-58

   1. Initial program 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in v around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. swap-sqr 99.4%

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

      5. unpow2 99.4%

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

      6. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. pow2 99.4%

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

   9. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -880000000000 \lor \neg \left(v \leq 2 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (3.0d0 + (2.0d0 / (r * r))) + ((((r * w) * (0.375d0 + (v * (-0.25d0)))) / ((v + (-1.0d0)) / (r * w))) - 4.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (3.0 + (2.0 / (r * r))) + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5);
end

Wolfram

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

Derivation

1. Initial program 87.6%

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

   1. associate--l- 87.6%

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

   2. associate-*l* 82.7%

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

   3. sqr-neg 82.7%

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

   4. associate-*l* 87.6%

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

   5. associate-/l* 89.3%

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

   6. fma-define 89.3%

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

3. Simplified 89.3%

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

   1. add-sqr-sqrt 89.2%

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

   2. *-un-lft-identity 89.2%

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

   3. times-frac 89.2%

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

   4. *-commutative 89.2%

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

   5. sqrt-prod 48.2%

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

   6. *-commutative 48.2%

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

   7. sqrt-prod 48.2%

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

   8. sqrt-prod 25.3%

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

   9. add-sqr-sqrt 38.0%

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

   10. associate-*r* 38.0%

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

   11. add-sqr-sqrt 72.2%

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

6. Applied egg-rr 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. associate-*r* 97.6%

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

   3. clear-num 97.3%

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

   4. un-div-inv 97.4%

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

   5. distribute-lft-in 97.4%

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

   6. metadata-eval 97.4%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\color{blue}{0.375} + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

   7. associate-*r* 97.4%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.375 + \color{blue}{\left(0.125 \cdot -2\right) \cdot v}\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

   8. metadata-eval 97.4%

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

8. Applied egg-rr 97.4%

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

9. Final simplification 97.4%

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

Alternative 11: 93.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Simplified 89.2%

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

5. Taylor expanded in v around 0 82.0%

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

   1. *-commutative 82.0%

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

   2. unpow2 82.0%

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

   3. unpow2 82.0%

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

   4. swap-sqr 94.7%

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

   5. unpow2 94.7%

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

   6. *-commutative 94.7%

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

7. Simplified 94.7%

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

   1. *-commutative 94.7%

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

   2. pow2 94.7%

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

9. Applied egg-rr 94.7%

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

10. Final simplification 94.7%

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

Alternative 12: 57.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Simplified 82.6%

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

5. Taylor expanded in r around 0 56.6%

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

6. Final simplification 56.6%

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

Alternative 13: 14.9% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 87.6%

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

3. Simplified 82.6%

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

5. Taylor expanded in r around 0 56.6%

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

6. Taylor expanded in r around inf 13.3%

   \[\leadsto \color{blue}{-1.5} \]

7. Final simplification 13.3%

   \[\leadsto -1.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))