Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.6%

Time: 16.6s

Alternatives: 11

Speedup: 1.5×

• 54.1% 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: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2 \cdot 10^{+102} \lor \neg \left(v \leq 102000000\right):\\ \;\;\;\;t_0 + \left(-1.5 - \frac{r \cdot w}{\frac{\frac{4}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - \left(0.125 \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot \left(w \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -2e+102) (not (<= v 102000000.0)))
     (+ t_0 (- -1.5 (/ (* r w) (/ (/ 4.0 r) w))))
     (+
      -4.5
      (-
       (+ 3.0 t_0)
       (* (* 0.125 (* r w)) (/ (* r (* w (+ 3.0 (* v -2.0)))) (- 1.0 v))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2e+102) || !(v <= 102000000.0)) {
		tmp = t_0 + (-1.5 - ((r * w) / ((4.0 / r) / w)));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - ((0.125 * (r * w)) * ((r * (w * (3.0 + (v * -2.0)))) / (1.0 - v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-2d+102)) .or. (.not. (v <= 102000000.0d0))) then
        tmp = t_0 + ((-1.5d0) - ((r * w) / ((4.0d0 / r) / w)))
    else
        tmp = (-4.5d0) + ((3.0d0 + t_0) - ((0.125d0 * (r * w)) * ((r * (w * (3.0d0 + (v * (-2.0d0))))) / (1.0d0 - v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2e+102) || !(v <= 102000000.0)) {
		tmp = t_0 + (-1.5 - ((r * w) / ((4.0 / r) / w)));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - ((0.125 * (r * w)) * ((r * (w * (3.0 + (v * -2.0)))) / (1.0 - v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2e+102) or not (v <= 102000000.0):
		tmp = t_0 + (-1.5 - ((r * w) / ((4.0 / r) / w)))
	else:
		tmp = -4.5 + ((3.0 + t_0) - ((0.125 * (r * w)) * ((r * (w * (3.0 + (v * -2.0)))) / (1.0 - v))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2e+102) || !(v <= 102000000.0))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) / Float64(Float64(4.0 / r) / w))));
	else
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(Float64(0.125 * Float64(r * w)) * Float64(Float64(r * Float64(w * Float64(3.0 + Float64(v * -2.0)))) / Float64(1.0 - v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2e+102) || ~((v <= 102000000.0)))
		tmp = t_0 + (-1.5 - ((r * w) / ((4.0 / r) / w)));
	else
		tmp = -4.5 + ((3.0 + t_0) - ((0.125 * (r * w)) * ((r * (w * (3.0 + (v * -2.0)))) / (1.0 - v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -2e+102], N[Not[LessEqual[v, 102000000.0]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] / N[(N[(4.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(0.125 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(w * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.99999999999999995e102 or 1.02e8 < v

   1. Initial program 85.0%

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

      1. associate--l- 85.0%

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

      2. +-commutative 85.0%

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

      3. associate--l+ 85.0%

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

      4. +-commutative 85.0%

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

      5. associate--r+ 85.0%

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

      6. metadata-eval 85.0%

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

      7. associate-*r* 82.3%

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

      8. *-commutative 82.3%

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

      9. associate-/l* 84.8%

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

      10. *-commutative 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in v around inf 90.0%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 90.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 90.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 94.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 94.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 94.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 94.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

      2. associate-/r* 94.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

      3. *-commutative 94.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \frac{\frac{4}{w}}{\color{blue}{w \cdot r}}}\right) \]

   8. Applied egg-rr 94.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 93.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)\right)}\right) \]

      2. expm1-udef 93.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)} - 1\right)}\right) \]

      3. *-un-lft-identity 93.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right)} - 1\right)\right) \]

      4. *-commutative 93.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right)} - 1\right)\right) \]

   10. Applied egg-rr 93.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)} - 1\right)}\right) \]
   11. Step-by-step derivation

       1. expm1-def 93.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\right)}\right) \]

       2. expm1-log1p 94.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

       3. associate-/r/ 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{4}{w}} \cdot \left(r \cdot w\right)}\right) \]

       4. associate-*l/ 97.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(r \cdot w\right)}{\frac{4}{w}}}\right) \]

       5. associate-*r/ 94.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{r \cdot \frac{r \cdot w}{\frac{4}{w}}}\right) \]

       6. *-commutative 94.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{\color{blue}{w \cdot r}}{\frac{4}{w}}\right) \]

       7. associate-/l* 94.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \color{blue}{\frac{w}{\frac{\frac{4}{w}}{r}}}\right) \]

       8. associate-/l/ 94.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\color{blue}{\frac{4}{r \cdot w}}}\right) \]

       9. associate-/r* 94.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\color{blue}{\frac{\frac{4}{r}}{w}}}\right) \]

   12. Simplified 94.7%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}}\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot w}{\frac{\frac{4}{r}}{w}}}\right) \]

   14. Applied egg-rr 99.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot w}{\frac{\frac{4}{r}}{w}}}\right) \]

   if -1.99999999999999995e102 < v < 1.02e8

   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 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. add-sqr-sqrt 79.5%

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

      3. times-frac 79.4%

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

      4. unswap-sqr 79.5%

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

      5. sqrt-prod 41.0%

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

      6. add-sqr-sqrt 62.7%

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

      7. unswap-sqr 78.5%

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

      8. sqrt-prod 50.5%

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

      9. add-sqr-sqrt 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. fma-def 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-/r/ 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r/ 99.7%

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

      5. *-commutative 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in r around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+102} \lor \neg \left(v \leq 102000000\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r \cdot w}{\frac{\frac{4}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot \left(w \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 86.8%

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

3. Simplified 81.9%

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

   1. *-un-lft-identity 81.9%

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

   2. add-sqr-sqrt 81.9%

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

   3. times-frac 81.9%

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

   4. unswap-sqr 81.9%

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

   5. sqrt-prod 43.8%

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

   6. add-sqr-sqrt 61.5%

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

   7. unswap-sqr 73.9%

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

   8. sqrt-prod 51.8%

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

   9. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. times-frac 99.8%

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

   2. +-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. fma-def 99.8%

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

7. Applied egg-rr 99.8%

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

   1. associate-/r/ 99.8%

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

   2. metadata-eval 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/r/ 99.8%

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

   5. *-commutative 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + v \cdot -2\\ t_1 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2 \cdot 10^{+105}:\\ \;\;\;\;-4.5 + \left(t_1 - \frac{0.125 \cdot t_0}{\frac{\frac{1 - v}{\left(r \cdot r\right) \cdot w}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(t_1 - \left(0.125 \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot \left(w \cdot t_0\right)}{1 - v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (* v -2.0))) (t_1 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= r 2e+105)
     (+ -4.5 (- t_1 (/ (* 0.125 t_0) (/ (/ (- 1.0 v) (* (* r r) w)) w))))
     (+ -4.5 (- t_1 (* (* 0.125 (* r w)) (/ (* r (* w t_0)) (- 1.0 v))))))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (v * -2.0);
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (r <= 2e+105) {
		tmp = -4.5 + (t_1 - ((0.125 * t_0) / (((1.0 - v) / ((r * r) * w)) / w)));
	} else {
		tmp = -4.5 + (t_1 - ((0.125 * (r * w)) * ((r * (w * t_0)) / (1.0 - v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + (v * (-2.0d0))
    t_1 = 3.0d0 + (2.0d0 / (r * r))
    if (r <= 2d+105) then
        tmp = (-4.5d0) + (t_1 - ((0.125d0 * t_0) / (((1.0d0 - v) / ((r * r) * w)) / w)))
    else
        tmp = (-4.5d0) + (t_1 - ((0.125d0 * (r * w)) * ((r * (w * t_0)) / (1.0d0 - v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (v * -2.0);
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (r <= 2e+105) {
		tmp = -4.5 + (t_1 - ((0.125 * t_0) / (((1.0 - v) / ((r * r) * w)) / w)));
	} else {
		tmp = -4.5 + (t_1 - ((0.125 * (r * w)) * ((r * (w * t_0)) / (1.0 - v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (v * -2.0)
	t_1 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if r <= 2e+105:
		tmp = -4.5 + (t_1 - ((0.125 * t_0) / (((1.0 - v) / ((r * r) * w)) / w)))
	else:
		tmp = -4.5 + (t_1 - ((0.125 * (r * w)) * ((r * (w * t_0)) / (1.0 - v))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(v * -2.0))
	t_1 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (r <= 2e+105)
		tmp = Float64(-4.5 + Float64(t_1 - Float64(Float64(0.125 * t_0) / Float64(Float64(Float64(1.0 - v) / Float64(Float64(r * r) * w)) / w))));
	else
		tmp = Float64(-4.5 + Float64(t_1 - Float64(Float64(0.125 * Float64(r * w)) * Float64(Float64(r * Float64(w * t_0)) / Float64(1.0 - v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (v * -2.0);
	t_1 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if (r <= 2e+105)
		tmp = -4.5 + (t_1 - ((0.125 * t_0) / (((1.0 - v) / ((r * r) * w)) / w)));
	else
		tmp = -4.5 + (t_1 - ((0.125 * (r * w)) * ((r * (w * t_0)) / (1.0 - v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2e+105], N[(-4.5 + N[(t$95$1 - N[(N[(0.125 * t$95$0), $MachinePrecision] / N[(N[(N[(1.0 - v), $MachinePrecision] / N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(t$95$1 - N[(N[(0.125 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(w * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.9999999999999999e105

   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 85.0%

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

      1. *-un-lft-identity 85.0%

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

      2. associate-*l* 94.0%

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

      3. times-frac 94.4%

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

   5. Applied egg-rr 94.4%

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

      1. associate-*l/ 94.4%

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

      2. *-un-lft-identity 94.4%

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

   7. Applied egg-rr 94.4%

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

   if 1.9999999999999999e105 < r

   1. Initial program 84.4%

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

   3. Simplified 67.3%

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

      1. *-un-lft-identity 67.3%

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

      2. add-sqr-sqrt 67.2%

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

      3. times-frac 67.2%

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

      4. unswap-sqr 67.2%

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

      5. sqrt-prod 35.8%

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

      6. add-sqr-sqrt 35.8%

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

      7. unswap-sqr 55.5%

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

      8. sqrt-prod 53.2%

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

      9. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. fma-def 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-/r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r/ 99.8%

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

      5. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in r around 0 95.6%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{+105}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + v \cdot -2\right)}{\frac{\frac{1 - v}{\left(r \cdot r\right) \cdot w}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot \left(w \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}\right)\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Simplified 81.9%

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

   1. *-un-lft-identity 81.9%

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

   2. add-sqr-sqrt 81.9%

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

   3. times-frac 81.9%

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

   4. unswap-sqr 81.9%

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

   5. sqrt-prod 43.8%

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

   6. add-sqr-sqrt 61.5%

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

   7. unswap-sqr 73.9%

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

   8. sqrt-prod 51.8%

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

   9. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2.75 \cdot 10^{+34} \lor \neg \left(v \leq 1.6 \cdot 10^{-27}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \frac{r \cdot w}{\frac{\frac{4}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -2.75e+34) (not (<= v 1.6e-27)))
     (+ t_0 (- -1.5 (/ (* r w) (/ (/ 4.0 r) w))))
     (+ t_0 (- -1.5 (* (* r w) (* w (* r 0.375))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2.75e+34) || !(v <= 1.6e-27)) {
		tmp = t_0 + (-1.5 - ((r * w) / ((4.0 / r) / w)));
	} else {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	}
	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 <= (-2.75d+34)) .or. (.not. (v <= 1.6d-27))) then
        tmp = t_0 + ((-1.5d0) - ((r * w) / ((4.0d0 / r) / w)))
    else
        tmp = t_0 + ((-1.5d0) - ((r * w) * (w * (r * 0.375d0))))
    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 <= -2.75e+34) || !(v <= 1.6e-27)) {
		tmp = t_0 + (-1.5 - ((r * w) / ((4.0 / r) / w)));
	} else {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2.75e+34) or not (v <= 1.6e-27):
		tmp = t_0 + (-1.5 - ((r * w) / ((4.0 / r) / w)))
	else:
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2.75e+34) || !(v <= 1.6e-27))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) / Float64(Float64(4.0 / r) / w))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2.75e+34) || ~((v <= 1.6e-27)))
		tmp = t_0 + (-1.5 - ((r * w) / ((4.0 / r) / w)));
	else
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	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, -2.75e+34], N[Not[LessEqual[v, 1.6e-27]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] / N[(N[(4.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.7499999999999998e34 or 1.59999999999999995e-27 < v

   1. Initial program 84.9%

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

      1. associate--l- 84.9%

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

      2. +-commutative 84.9%

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

      3. associate--l+ 84.9%

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

      4. +-commutative 84.9%

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

      5. associate--r+ 84.9%

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

      6. metadata-eval 84.9%

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

      7. associate-*r* 82.7%

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

      8. *-commutative 82.7%

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

      9. associate-/l* 85.5%

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

      10. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in v around inf 89.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 89.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 89.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 94.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 94.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 94.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 94.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

      2. associate-/r* 94.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

      3. *-commutative 94.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \frac{\frac{4}{w}}{\color{blue}{w \cdot r}}}\right) \]

   8. Applied egg-rr 94.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 93.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)\right)}\right) \]

      2. expm1-udef 93.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)} - 1\right)}\right) \]

      3. *-un-lft-identity 93.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right)} - 1\right)\right) \]

      4. *-commutative 93.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right)} - 1\right)\right) \]

   10. Applied egg-rr 93.4%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)} - 1\right)}\right) \]
   11. Step-by-step derivation

       1. expm1-def 93.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\right)}\right) \]

       2. expm1-log1p 94.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

       3. associate-/r/ 99.5%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{4}{w}} \cdot \left(r \cdot w\right)}\right) \]

       4. associate-*l/ 97.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(r \cdot w\right)}{\frac{4}{w}}}\right) \]

       5. associate-*r/ 94.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{r \cdot \frac{r \cdot w}{\frac{4}{w}}}\right) \]

       6. *-commutative 94.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{\color{blue}{w \cdot r}}{\frac{4}{w}}\right) \]

       7. associate-/l* 94.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \color{blue}{\frac{w}{\frac{\frac{4}{w}}{r}}}\right) \]

       8. associate-/l/ 94.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\color{blue}{\frac{4}{r \cdot w}}}\right) \]

       9. associate-/r* 94.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\color{blue}{\frac{\frac{4}{r}}{w}}}\right) \]

   12. Simplified 94.3%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}}\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 99.5%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot w}{\frac{\frac{4}{r}}{w}}}\right) \]

   14. Applied egg-rr 99.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot w}{\frac{\frac{4}{r}}{w}}}\right) \]

   if -2.7499999999999998e34 < v < 1.59999999999999995e-27

   1. Initial program 88.7%

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

      1. associate--l- 88.7%

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

      2. +-commutative 88.7%

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

      3. associate--l+ 88.7%

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

      4. +-commutative 88.7%

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

      5. associate--r+ 88.7%

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

      6. metadata-eval 88.7%

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

      7. associate-*r* 88.7%

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

      8. *-commutative 88.7%

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

      9. associate-/l* 88.7%

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

      10. *-commutative 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in v around 0 88.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 88.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 88.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 98.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 98.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 98.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 98.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

      2. associate-*r* 99.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\frac{r}{2.6666666666666665} \cdot w\right) \cdot \left(r \cdot w\right)}\right) \]

      3. div-inv 99.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{\left(r \cdot \frac{1}{2.6666666666666665}\right)} \cdot w\right) \cdot \left(r \cdot w\right)\right) \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.75 \cdot 10^{+34} \lor \neg \left(v \leq 1.6 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r \cdot w}{\frac{\frac{4}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \end{array} \]

Alternative 6: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2.75 \cdot 10^{+34}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot 0.25\right)\right)\\ \mathbf{elif}\;v \leq 10^{-19}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -2.75e+34)
     (+ t_0 (- -1.5 (* (* r (* r w)) (* w 0.25))))
     (if (<= v 1e-19)
       (+ t_0 (- -1.5 (* (* r w) (* w (* r 0.375)))))
       (+ t_0 (- -1.5 (* r (/ w (/ (/ 4.0 r) w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -2.75e+34) {
		tmp = t_0 + (-1.5 - ((r * (r * w)) * (w * 0.25)));
	} else if (v <= 1e-19) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	} else {
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / 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 <= (-2.75d+34)) then
        tmp = t_0 + ((-1.5d0) - ((r * (r * w)) * (w * 0.25d0)))
    else if (v <= 1d-19) then
        tmp = t_0 + ((-1.5d0) - ((r * w) * (w * (r * 0.375d0))))
    else
        tmp = t_0 + ((-1.5d0) - (r * (w / ((4.0d0 / 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 <= -2.75e+34) {
		tmp = t_0 + (-1.5 - ((r * (r * w)) * (w * 0.25)));
	} else if (v <= 1e-19) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	} else {
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / r) / w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -2.75e+34:
		tmp = t_0 + (-1.5 - ((r * (r * w)) * (w * 0.25)))
	elif v <= 1e-19:
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))))
	else:
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / r) / w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -2.75e+34)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(r * w)) * Float64(w * 0.25))));
	elseif (v <= 1e-19)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(r * Float64(w / Float64(Float64(4.0 / 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 <= -2.75e+34)
		tmp = t_0 + (-1.5 - ((r * (r * w)) * (w * 0.25)));
	elseif (v <= 1e-19)
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	else
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / 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[LessEqual[v, -2.75e+34], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1e-19], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(r * N[(w / N[(N[(4.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.7499999999999998e34

   1. Initial program 84.3%

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

      1. associate--l- 84.3%

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

      2. +-commutative 84.3%

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

      3. associate--l+ 84.3%

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

      4. +-commutative 84.3%

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

      5. associate--r+ 84.3%

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

      6. metadata-eval 84.3%

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

      7. associate-*r* 84.3%

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

      8. *-commutative 84.3%

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

      9. associate-/l* 86.0%

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

      10. *-commutative 86.0%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in v around inf 87.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 87.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 87.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 92.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

      2. associate-/r* 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

      3. *-commutative 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \frac{\frac{4}{w}}{\color{blue}{w \cdot r}}}\right) \]

   8. Applied egg-rr 92.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}}\right) \]
   9. Step-by-step derivation

      1. *-un-lft-identity 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{1 \cdot r}}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right) \]

      2. *-un-lft-identity 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1 \cdot r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

      3. div-inv 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1 \cdot r}{\color{blue}{\frac{4}{w} \cdot \frac{1}{w \cdot r}}}\right) \]

      4. *-commutative 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1 \cdot r}{\frac{4}{w} \cdot \frac{1}{\color{blue}{r \cdot w}}}\right) \]

      5. times-frac 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{1}{\frac{4}{w}} \cdot \frac{r}{\frac{1}{r \cdot w}}}\right) \]

      6. clear-num 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{w}{4}} \cdot \frac{r}{\frac{1}{r \cdot w}}\right) \]

      7. div-inv 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(w \cdot \frac{1}{4}\right)} \cdot \frac{r}{\frac{1}{r \cdot w}}\right) \]

      8. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-commutative 99.9%

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

       2. associate-/r/ 99.9%

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

       3. /-rgt-identity 99.9%

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

   12. Simplified 99.9%

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

   if -2.7499999999999998e34 < v < 9.9999999999999998e-20

   1. Initial program 88.1%

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

      1. associate--l- 88.0%

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

      2. +-commutative 88.0%

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

      3. associate--l+ 88.0%

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

      4. +-commutative 88.0%

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

      5. associate--r+ 88.1%

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

      6. metadata-eval 88.1%

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

      7. associate-*r* 88.1%

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

      8. *-commutative 88.1%

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

      9. associate-/l* 88.1%

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

      10. *-commutative 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in v around 0 87.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 87.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 87.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 97.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 97.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 97.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 97.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

      2. associate-*r* 99.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\frac{r}{2.6666666666666665} \cdot w\right) \cdot \left(r \cdot w\right)}\right) \]

      3. div-inv 99.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{\left(r \cdot \frac{1}{2.6666666666666665}\right)} \cdot w\right) \cdot \left(r \cdot w\right)\right) \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

   if 9.9999999999999998e-20 < v

   1. Initial program 86.3%

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

      1. associate--l- 86.3%

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

      2. +-commutative 86.3%

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

      3. associate--l+ 86.3%

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

      4. +-commutative 86.3%

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

      5. associate--r+ 86.3%

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

      6. metadata-eval 86.3%

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

      7. associate-*r* 82.5%

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

      8. *-commutative 82.5%

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

      9. associate-/l* 86.1%

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

      10. *-commutative 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in v around inf 91.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 91.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 91.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 96.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 96.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 96.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 96.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

      2. associate-/r* 96.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

      3. *-commutative 96.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \frac{\frac{4}{w}}{\color{blue}{w \cdot r}}}\right) \]

   8. Applied egg-rr 96.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)\right)}\right) \]

      2. expm1-udef 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)} - 1\right)}\right) \]

      3. *-un-lft-identity 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right)} - 1\right)\right) \]

      4. *-commutative 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right)} - 1\right)\right) \]

   10. Applied egg-rr 95.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)} - 1\right)}\right) \]
   11. Step-by-step derivation

       1. expm1-def 95.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\right)}\right) \]

       2. expm1-log1p 96.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

       3. associate-/r/ 99.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{4}{w}} \cdot \left(r \cdot w\right)}\right) \]

       4. associate-*l/ 95.5%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(r \cdot w\right)}{\frac{4}{w}}}\right) \]

       5. associate-*r/ 96.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{r \cdot \frac{r \cdot w}{\frac{4}{w}}}\right) \]

       6. *-commutative 96.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{\color{blue}{w \cdot r}}{\frac{4}{w}}\right) \]

       7. associate-/l* 96.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \color{blue}{\frac{w}{\frac{\frac{4}{w}}{r}}}\right) \]

       8. associate-/l/ 96.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\color{blue}{\frac{4}{r \cdot w}}}\right) \]

       9. associate-/r* 96.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\color{blue}{\frac{\frac{4}{r}}{w}}}\right) \]

   12. Simplified 96.7%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.75 \cdot 10^{+34}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot 0.25\right)\right)\\ \mathbf{elif}\;v \leq 10^{-19}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}\right)\\ \end{array} \]

Alternative 7: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3.35 \cdot 10^{+34}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot 0.25\right)\right)\\ \mathbf{elif}\;v \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r}{\frac{4}{w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -3.35e+34)
     (+ t_0 (- -1.5 (* (* r (* r w)) (* w 0.25))))
     (if (<= v 1.6e-27)
       (+ t_0 (- -1.5 (* (* r w) (* w (* r 0.375)))))
       (+ t_0 (- -1.5 (* (* r w) (/ r (/ 4.0 w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -3.35e+34) {
		tmp = t_0 + (-1.5 - ((r * (r * w)) * (w * 0.25)));
	} else if (v <= 1.6e-27) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	} else {
		tmp = t_0 + (-1.5 - ((r * w) * (r / (4.0 / 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 <= (-3.35d+34)) then
        tmp = t_0 + ((-1.5d0) - ((r * (r * w)) * (w * 0.25d0)))
    else if (v <= 1.6d-27) then
        tmp = t_0 + ((-1.5d0) - ((r * w) * (w * (r * 0.375d0))))
    else
        tmp = t_0 + ((-1.5d0) - ((r * w) * (r / (4.0d0 / 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 <= -3.35e+34) {
		tmp = t_0 + (-1.5 - ((r * (r * w)) * (w * 0.25)));
	} else if (v <= 1.6e-27) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	} else {
		tmp = t_0 + (-1.5 - ((r * w) * (r / (4.0 / w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -3.35e+34:
		tmp = t_0 + (-1.5 - ((r * (r * w)) * (w * 0.25)))
	elif v <= 1.6e-27:
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))))
	else:
		tmp = t_0 + (-1.5 - ((r * w) * (r / (4.0 / w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -3.35e+34)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(r * w)) * Float64(w * 0.25))));
	elseif (v <= 1.6e-27)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(r / Float64(4.0 / w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -3.35e+34)
		tmp = t_0 + (-1.5 - ((r * (r * w)) * (w * 0.25)));
	elseif (v <= 1.6e-27)
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	else
		tmp = t_0 + (-1.5 - ((r * w) * (r / (4.0 / 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, -3.35e+34], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1.6e-27], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(r / N[(4.0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.3500000000000001e34

   1. Initial program 84.3%

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

      1. associate--l- 84.3%

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

      2. +-commutative 84.3%

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

      3. associate--l+ 84.3%

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

      4. +-commutative 84.3%

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

      5. associate--r+ 84.3%

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

      6. metadata-eval 84.3%

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

      7. associate-*r* 84.3%

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

      8. *-commutative 84.3%

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

      9. associate-/l* 86.0%

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

      10. *-commutative 86.0%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in v around inf 87.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 87.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 87.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 92.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

      2. associate-/r* 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

      3. *-commutative 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \frac{\frac{4}{w}}{\color{blue}{w \cdot r}}}\right) \]

   8. Applied egg-rr 92.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}}\right) \]
   9. Step-by-step derivation

      1. *-un-lft-identity 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{1 \cdot r}}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right) \]

      2. *-un-lft-identity 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1 \cdot r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

      3. div-inv 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1 \cdot r}{\color{blue}{\frac{4}{w} \cdot \frac{1}{w \cdot r}}}\right) \]

      4. *-commutative 92.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1 \cdot r}{\frac{4}{w} \cdot \frac{1}{\color{blue}{r \cdot w}}}\right) \]

      5. times-frac 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{1}{\frac{4}{w}} \cdot \frac{r}{\frac{1}{r \cdot w}}}\right) \]

      6. clear-num 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{w}{4}} \cdot \frac{r}{\frac{1}{r \cdot w}}\right) \]

      7. div-inv 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(w \cdot \frac{1}{4}\right)} \cdot \frac{r}{\frac{1}{r \cdot w}}\right) \]

      8. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-commutative 99.9%

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

       2. associate-/r/ 99.9%

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

       3. /-rgt-identity 99.9%

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

   12. Simplified 99.9%

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

   if -3.3500000000000001e34 < v < 1.59999999999999995e-27

   1. Initial program 88.7%

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

      1. associate--l- 88.7%

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

      2. +-commutative 88.7%

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

      3. associate--l+ 88.7%

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

      4. +-commutative 88.7%

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

      5. associate--r+ 88.7%

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

      6. metadata-eval 88.7%

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

      7. associate-*r* 88.7%

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

      8. *-commutative 88.7%

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

      9. associate-/l* 88.7%

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

      10. *-commutative 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in v around 0 88.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 88.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 88.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 98.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 98.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 98.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 98.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

      2. associate-*r* 99.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\frac{r}{2.6666666666666665} \cdot w\right) \cdot \left(r \cdot w\right)}\right) \]

      3. div-inv 99.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{\left(r \cdot \frac{1}{2.6666666666666665}\right)} \cdot w\right) \cdot \left(r \cdot w\right)\right) \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

   if 1.59999999999999995e-27 < v

   1. Initial program 85.4%

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

      1. associate--l- 85.4%

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

      2. +-commutative 85.4%

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

      3. associate--l+ 85.4%

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

      4. +-commutative 85.4%

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

      5. associate--r+ 85.4%

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

      6. metadata-eval 85.4%

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

      7. associate-*r* 81.7%

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

      8. *-commutative 81.7%

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

      9. associate-/l* 85.2%

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

      10. *-commutative 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in v around inf 90.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 90.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 90.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 95.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 95.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 95.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 95.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

      2. associate-/r* 95.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

      3. *-commutative 95.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \frac{\frac{4}{w}}{\color{blue}{w \cdot r}}}\right) \]

   8. Applied egg-rr 95.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 94.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)\right)}\right) \]

      2. expm1-udef 94.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)} - 1\right)}\right) \]

      3. *-un-lft-identity 94.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right)} - 1\right)\right) \]

      4. *-commutative 94.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right)} - 1\right)\right) \]

   10. Applied egg-rr 94.7%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)} - 1\right)}\right) \]
   11. Step-by-step derivation

       1. expm1-def 94.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\right)}\right) \]

       2. expm1-log1p 95.5%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

       3. associate-/r/ 99.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{4}{w}} \cdot \left(r \cdot w\right)}\right) \]

   12. Simplified 99.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{4}{w}} \cdot \left(r \cdot w\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.35 \cdot 10^{+34}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot 0.25\right)\right)\\ \mathbf{elif}\;v \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r}{\frac{4}{w}}\right)\\ \end{array} \]

Alternative 8: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 5.5 \cdot 10^{-216}:\\ \;\;\;\;t_0 + -1.5\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 5.5e-216)
     (+ t_0 -1.5)
     (+ t_0 (* (* (* r r) (* w w)) -0.375)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 5.5e-216) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + (((r * r) * (w * w)) * -0.375);
	}
	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 ((w * w) <= 5.5d-216) then
        tmp = t_0 + (-1.5d0)
    else
        tmp = t_0 + (((r * r) * (w * w)) * (-0.375d0))
    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 ((w * w) <= 5.5e-216) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + (((r * r) * (w * w)) * -0.375);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 5.5e-216:
		tmp = t_0 + -1.5
	else:
		tmp = t_0 + (((r * r) * (w * w)) * -0.375)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 5.5e-216)
		tmp = Float64(t_0 + -1.5);
	else
		tmp = Float64(t_0 + Float64(Float64(Float64(r * r) * Float64(w * w)) * -0.375));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 5.5e-216)
		tmp = t_0 + -1.5;
	else
		tmp = t_0 + (((r * r) * (w * w)) * -0.375);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 5.5e-216], N[(t$95$0 + -1.5), $MachinePrecision], N[(t$95$0 + N[(N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 5.49999999999999991e-216

   1. Initial program 89.6%

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

      1. associate--l- 89.6%

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

      2. +-commutative 89.6%

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

      3. associate--l+ 89.6%

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

      4. +-commutative 89.6%

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

      5. associate--r+ 89.6%

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

      6. metadata-eval 89.6%

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

      7. associate-*r* 89.5%

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

      8. *-commutative 89.5%

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

      9. associate-/l* 90.7%

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

      10. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in v around 0 88.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 88.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 88.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 93.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

      2. associate-*r* 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\frac{r}{2.6666666666666665} \cdot w\right) \cdot \left(r \cdot w\right)}\right) \]

      3. div-inv 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{\left(r \cdot \frac{1}{2.6666666666666665}\right)} \cdot w\right) \cdot \left(r \cdot w\right)\right) \]

      4. metadata-eval 93.3%

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

   8. Applied egg-rr 93.3%

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

   9. Taylor expanded in r around 0 84.8%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   if 5.49999999999999991e-216 < (*.f64 w w)

   1. Initial program 85.5%

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

      1. associate--l- 85.5%

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

      2. +-commutative 85.5%

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

      3. associate--l+ 85.5%

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

      4. +-commutative 85.5%

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

      5. associate--r+ 85.5%

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

      6. metadata-eval 85.5%

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

      7. associate-*r* 83.8%

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

      8. *-commutative 83.8%

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

      9. associate-/l* 85.4%

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

      10. *-commutative 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in v around 0 83.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 83.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 83.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 90.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 90.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 90.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 90.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

      2. associate-*r* 93.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\frac{r}{2.6666666666666665} \cdot w\right) \cdot \left(r \cdot w\right)}\right) \]

      3. div-inv 93.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{\left(r \cdot \frac{1}{2.6666666666666665}\right)} \cdot w\right) \cdot \left(r \cdot w\right)\right) \]

      4. metadata-eval 93.5%

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

   8. Applied egg-rr 93.5%

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

   9. Taylor expanded in r around inf 80.5%

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

       1. *-commutative 80.5%

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

       2. unpow2 80.5%

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

       3. unpow2 80.5%

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

   11. Simplified 80.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 5.5 \cdot 10^{-216}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\\ \end{array} \]

Alternative 9: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2 \cdot 10^{-149}:\\ \;\;\;\;t_0 + -1.5\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2e-149)
     (+ t_0 -1.5)
     (+ t_0 (- -1.5 (* r (/ w (/ (/ 4.0 r) w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2e-149) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / 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 (r <= 2d-149) then
        tmp = t_0 + (-1.5d0)
    else
        tmp = t_0 + ((-1.5d0) - (r * (w / ((4.0d0 / 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 (r <= 2e-149) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / r) / w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2e-149:
		tmp = t_0 + -1.5
	else:
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / r) / w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2e-149)
		tmp = Float64(t_0 + -1.5);
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(r * Float64(w / Float64(Float64(4.0 / r) / w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2e-149)
		tmp = t_0 + -1.5;
	else
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / 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[LessEqual[r, 2e-149], N[(t$95$0 + -1.5), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(r * N[(w / N[(N[(4.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.99999999999999996e-149

   1. Initial program 84.5%

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

      1. associate--l- 84.5%

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

      2. +-commutative 84.5%

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

      3. associate--l+ 84.5%

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

      4. +-commutative 84.5%

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

      5. associate--r+ 84.5%

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

      6. metadata-eval 84.5%

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

      7. associate-*r* 83.2%

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

      8. *-commutative 83.2%

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

      9. associate-/l* 83.8%

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

      10. *-commutative 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in v around 0 83.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 83.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 83.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 92.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 92.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 92.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 92.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

      2. associate-*r* 95.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\frac{r}{2.6666666666666665} \cdot w\right) \cdot \left(r \cdot w\right)}\right) \]

      3. div-inv 95.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{\left(r \cdot \frac{1}{2.6666666666666665}\right)} \cdot w\right) \cdot \left(r \cdot w\right)\right) \]

      4. metadata-eval 95.4%

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

   8. Applied egg-rr 95.4%

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

   9. Taylor expanded in r around 0 62.8%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   if 1.99999999999999996e-149 < r

   1. Initial program 90.6%

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

      1. associate--l- 90.6%

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

      2. +-commutative 90.6%

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

      3. associate--l+ 90.6%

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

      4. +-commutative 90.6%

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

      5. associate--r+ 90.6%

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

      6. metadata-eval 90.6%

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

      7. associate-*r* 89.6%

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

      8. *-commutative 89.6%

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

      9. associate-/l* 92.5%

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

      10. *-commutative 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in v around inf 89.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 89.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 89.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 92.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 92.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 92.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 92.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

      2. associate-/r* 92.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

      3. *-commutative 92.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \frac{\frac{4}{w}}{\color{blue}{w \cdot r}}}\right) \]

   8. Applied egg-rr 92.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 92.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)\right)}\right) \]

      2. expm1-udef 92.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)} - 1\right)}\right) \]

      3. *-un-lft-identity 92.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right)} - 1\right)\right) \]

      4. *-commutative 92.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right)} - 1\right)\right) \]

   10. Applied egg-rr 92.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)} - 1\right)}\right) \]
   11. Step-by-step derivation

       1. expm1-def 92.0%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\right)}\right) \]

       2. expm1-log1p 92.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

       3. associate-/r/ 92.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{4}{w}} \cdot \left(r \cdot w\right)}\right) \]

       4. associate-*l/ 90.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(r \cdot w\right)}{\frac{4}{w}}}\right) \]

       5. associate-*r/ 92.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{r \cdot \frac{r \cdot w}{\frac{4}{w}}}\right) \]

       6. *-commutative 92.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{\color{blue}{w \cdot r}}{\frac{4}{w}}\right) \]

       7. associate-/l* 92.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \color{blue}{\frac{w}{\frac{\frac{4}{w}}{r}}}\right) \]

       8. associate-/l/ 92.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\color{blue}{\frac{4}{r \cdot w}}}\right) \]

       9. associate-/r* 92.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\color{blue}{\frac{\frac{4}{r}}{w}}}\right) \]

   12. Simplified 92.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}\right)\\ \end{array} \]

Alternative 10: 90.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 6.2 \cdot 10^{+140}:\\ \;\;\;\;t_0 + \left(-1.5 - w \cdot \left(\left(\left(r \cdot r\right) \cdot w\right) \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 6.2e+140)
     (+ t_0 (- -1.5 (* w (* (* (* r r) w) 0.375))))
     (+ t_0 (- -1.5 (* r (/ w (/ (/ 4.0 r) w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.2e+140) {
		tmp = t_0 + (-1.5 - (w * (((r * r) * w) * 0.375)));
	} else {
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / 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 (r <= 6.2d+140) then
        tmp = t_0 + ((-1.5d0) - (w * (((r * r) * w) * 0.375d0)))
    else
        tmp = t_0 + ((-1.5d0) - (r * (w / ((4.0d0 / 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 (r <= 6.2e+140) {
		tmp = t_0 + (-1.5 - (w * (((r * r) * w) * 0.375)));
	} else {
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / r) / w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 6.2e+140:
		tmp = t_0 + (-1.5 - (w * (((r * r) * w) * 0.375)))
	else:
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / r) / w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 6.2e+140)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(w * Float64(Float64(Float64(r * r) * w) * 0.375))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(r * Float64(w / Float64(Float64(4.0 / r) / w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 6.2e+140)
		tmp = t_0 + (-1.5 - (w * (((r * r) * w) * 0.375)));
	else
		tmp = t_0 + (-1.5 - (r * (w / ((4.0 / 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[LessEqual[r, 6.2e+140], N[(t$95$0 + N[(-1.5 - N[(w * N[(N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(r * N[(w / N[(N[(4.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 6.2000000000000001e140

   1. Initial program 87.8%

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

      1. associate--l- 87.8%

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

      2. +-commutative 87.8%

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

      3. associate--l+ 87.8%

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

      4. +-commutative 87.8%

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

      5. associate--r+ 87.8%

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

      6. metadata-eval 87.8%

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

      7. associate-*r* 86.4%

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

      8. *-commutative 86.4%

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

      9. associate-/l* 86.9%

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

      10. *-commutative 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in v around 0 86.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 86.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 86.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 92.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 92.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 92.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r* 92.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{2.6666666666666665}{w}}{r \cdot w}}}\right) \]

      2. associate-/r/ 95.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{2.6666666666666665}{w}} \cdot \left(r \cdot w\right)}\right) \]

      3. associate-*r* 93.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\frac{r}{\frac{2.6666666666666665}{w}} \cdot r\right) \cdot w}\right) \]

      4. div-inv 93.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{\left(r \cdot \frac{1}{\frac{2.6666666666666665}{w}}\right)} \cdot r\right) \cdot w\right) \]

      5. clear-num 93.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot \color{blue}{\frac{w}{2.6666666666666665}}\right) \cdot r\right) \cdot w\right) \]

      6. div-inv 93.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot \color{blue}{\left(w \cdot \frac{1}{2.6666666666666665}\right)}\right) \cdot r\right) \cdot w\right) \]

      7. metadata-eval 93.8%

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

   8. Applied egg-rr 93.8%

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

   9. Taylor expanded in r around 0 90.7%

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

       1. *-commutative 90.7%

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

       2. unpow2 90.7%

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

   11. Simplified 90.7%

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

   if 6.2000000000000001e140 < r

   1. Initial program 80.6%

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

      1. associate--l- 80.6%

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

      2. +-commutative 80.6%

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

      3. associate--l+ 80.6%

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

      4. +-commutative 80.6%

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

      5. associate--r+ 80.6%

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

      6. metadata-eval 80.6%

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

      7. associate-*r* 80.6%

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

      8. *-commutative 80.6%

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

      9. associate-/l* 88.3%

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

      10. *-commutative 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in v around inf 83.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 83.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 83.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 93.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

      2. associate-/r* 93.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

      3. *-commutative 93.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{1 \cdot \frac{\frac{4}{w}}{\color{blue}{w \cdot r}}}\right) \]

   8. Applied egg-rr 93.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 91.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)\right)}\right) \]

      2. expm1-udef 91.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{1 \cdot \frac{\frac{4}{w}}{w \cdot r}}\right)} - 1\right)}\right) \]

      3. *-un-lft-identity 91.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right)} - 1\right)\right) \]

      4. *-commutative 91.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right)} - 1\right)\right) \]

   10. Applied egg-rr 91.7%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)} - 1\right)}\right) \]
   11. Step-by-step derivation

       1. expm1-def 91.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\right)}\right) \]

       2. expm1-log1p 93.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

       3. associate-/r/ 93.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{4}{w}} \cdot \left(r \cdot w\right)}\right) \]

       4. associate-*l/ 88.0%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(r \cdot w\right)}{\frac{4}{w}}}\right) \]

       5. associate-*r/ 93.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{r \cdot \frac{r \cdot w}{\frac{4}{w}}}\right) \]

       6. *-commutative 93.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{\color{blue}{w \cdot r}}{\frac{4}{w}}\right) \]

       7. associate-/l* 93.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \color{blue}{\frac{w}{\frac{\frac{4}{w}}{r}}}\right) \]

       8. associate-/l/ 93.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\color{blue}{\frac{4}{r \cdot w}}}\right) \]

       9. associate-/r* 93.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\color{blue}{\frac{\frac{4}{r}}{w}}}\right) \]

   12. Simplified 93.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - w \cdot \left(\left(\left(r \cdot r\right) \cdot w\right) \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - r \cdot \frac{w}{\frac{\frac{4}{r}}{w}}\right)\\ \end{array} \]

Alternative 11: 57.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -1.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. associate--l- 86.8%

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

   2. +-commutative 86.8%

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

   3. associate--l+ 86.8%

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

   4. +-commutative 86.8%

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

   5. associate--r+ 86.8%

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

   6. metadata-eval 86.8%

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

   7. associate-*r* 85.6%

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

   8. *-commutative 85.6%

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

   9. associate-/l* 87.1%

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

   10. *-commutative 87.1%

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

3. Simplified 87.1%

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

4. Taylor expanded in v around 0 85.3%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
5. Step-by-step derivation

   1. unpow2 85.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

   2. *-commutative 85.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

   3. associate-*l* 91.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

   4. *-commutative 91.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

6. Simplified 91.4%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{w \cdot \left(r \cdot w\right)}}}\right) \]
7. Step-by-step derivation

   1. associate-/r/ 91.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

   2. associate-*r* 93.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\frac{r}{2.6666666666666665} \cdot w\right) \cdot \left(r \cdot w\right)}\right) \]

   3. div-inv 93.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{\left(r \cdot \frac{1}{2.6666666666666665}\right)} \cdot w\right) \cdot \left(r \cdot w\right)\right) \]

   4. metadata-eval 93.4%

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

8. Applied egg-rr 93.4%

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

9. Taylor expanded in r around 0 54.0%

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

10. Final simplification 54.0%

    \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

Reproduce

herbie shell --seed 2023282 
(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))