Rosa's TurbineBenchmark

Percentage Accurate: 85.1% → 95.9%

Time: 13.5s

Alternatives: 9

Speedup: 1.7×

• 54.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% 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: 95.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 7 \cdot 10^{-65}:\\ \;\;\;\;t_1 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{elif}\;v \leq 1.95 \cdot 10^{+270}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ (/ 2.0 r) r) (- -1.5 (/ r (/ 4.0 (* w (* r w)))))))
        (t_1 (/ 2.0 (* r r))))
   (if (<= v -2e-13)
     t_0
     (if (<= v 7e-65)
       (+ t_1 (- -1.5 (* (* r w) (* w (* r 0.375)))))
       (if (<= v 1.95e+270) t_0 (+ t_1 -1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -2e-13) {
		tmp = t_0;
	} else if (v <= 7e-65) {
		tmp = t_1 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	} else if (v <= 1.95e+270) {
		tmp = t_0;
	} else {
		tmp = t_1 + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((2.0d0 / r) / r) + ((-1.5d0) - (r / (4.0d0 / (w * (r * w)))))
    t_1 = 2.0d0 / (r * r)
    if (v <= (-2d-13)) then
        tmp = t_0
    else if (v <= 7d-65) then
        tmp = t_1 + ((-1.5d0) - ((r * w) * (w * (r * 0.375d0))))
    else if (v <= 1.95d+270) then
        tmp = t_0
    else
        tmp = t_1 + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -2e-13) {
		tmp = t_0;
	} else if (v <= 7e-65) {
		tmp = t_1 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	} else if (v <= 1.95e+270) {
		tmp = t_0;
	} else {
		tmp = t_1 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))))
	t_1 = 2.0 / (r * r)
	tmp = 0
	if v <= -2e-13:
		tmp = t_0
	elif v <= 7e-65:
		tmp = t_1 + (-1.5 - ((r * w) * (w * (r * 0.375))))
	elif v <= 1.95e+270:
		tmp = t_0
	else:
		tmp = t_1 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(r / Float64(4.0 / Float64(w * Float64(r * w))))))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -2e-13)
		tmp = t_0;
	elseif (v <= 7e-65)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))));
	elseif (v <= 1.95e+270)
		tmp = t_0;
	else
		tmp = Float64(t_1 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -2e-13)
		tmp = t_0;
	elseif (v <= 7e-65)
		tmp = t_1 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	elseif (v <= 1.95e+270)
		tmp = t_0;
	else
		tmp = t_1 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(r / N[(4.0 / N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2e-13], t$95$0, If[LessEqual[v, 7e-65], N[(t$95$1 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1.95e+270], t$95$0, N[(t$95$1 + -1.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.0000000000000001e-13 or 7.00000000000000009e-65 < v < 1.95e270

   1. Initial program 85.2%

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

      1. associate--l- 85.2%

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

      2. +-commutative 85.2%

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

      3. associate--l+ 85.2%

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

      4. +-commutative 85.2%

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

      5. associate--r+ 85.2%

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

      6. metadata-eval 85.2%

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

      7. associate-*r* 84.5%

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

      8. *-commutative 84.5%

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

      9. associate-/l* 86.5%

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

      10. *-commutative 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in v around inf 89.9%

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

      1. unpow2 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.8%

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

   10. Simplified 99.8%

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

       1. inv-pow 99.8%

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

       2. associate-/r/ 99.7%

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

       3. unpow-prod-down 99.8%

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

       4. inv-pow 99.8%

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

       5. clear-num 99.8%

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

       6. inv-pow 99.8%

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

   12. Applied egg-rr 99.8%

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

       1. un-div-inv 99.8%

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

   14. Applied egg-rr 99.8%

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

   if -2.0000000000000001e-13 < v < 7.00000000000000009e-65

   1. Initial program 84.7%

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

      1. associate--l- 84.7%

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

      2. +-commutative 84.7%

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

      3. associate--l+ 84.7%

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

      4. +-commutative 84.7%

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

      5. associate--r+ 84.7%

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

      6. metadata-eval 84.7%

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

      7. associate-*r* 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-/l* 84.6%

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

      10. *-commutative 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in v around 0 84.6%

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

      1. unpow2 84.6%

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

      2. *-commutative 84.6%

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

      3. associate-*l* 95.2%

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

      4. *-commutative 95.2%

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

   6. Simplified 95.2%

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

      1. associate-/r/ 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*r* 99.7%

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

      4. div-inv 99.7%

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

      5. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 1.95e270 < v

   1. Initial program 66.7%

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

      1. associate--l- 66.7%

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

      2. +-commutative 66.7%

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

      3. associate--l+ 66.7%

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

      4. +-commutative 66.7%

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

      5. associate--r+ 66.7%

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

      6. metadata-eval 66.7%

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

      7. associate-*r* 66.7%

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

      8. *-commutative 66.7%

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

      9. associate-/l* 66.7%

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

      10. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in v around 0 66.7%

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

      1. unpow2 66.7%

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

      2. *-commutative 66.7%

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

      3. associate-*l* 88.9%

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

      4. *-commutative 88.9%

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

   6. Simplified 88.9%

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

      1. associate-/r/ 88.9%

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

      2. *-commutative 88.9%

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

      3. associate-*r* 100.0%

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

      4. div-inv 100.0%

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

      5. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in r around 0 100.0%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   10. Step-by-step derivation

       1. sub-neg 100.0%

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

       2. metadata-eval 100.0%

          \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{-1.5} \]

       3. unpow2 100.0%

          \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

       4. associate-*r/ 100.0%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{r \cdot r}} + -1.5 \]

       5. metadata-eval 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{2}{r \cdot r} + -1.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{elif}\;v \leq 7 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{elif}\;v \leq 1.95 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

3. Simplified 83.3%

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

   1. *-un-lft-identity 83.3%

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

   2. add-sqr-sqrt 83.2%

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

   3. times-frac 83.2%

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

   4. unswap-sqr 83.2%

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

   5. sqrt-prod 50.4%

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

   6. add-sqr-sqrt 64.1%

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

   7. unswap-sqr 78.8%

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

   8. sqrt-prod 58.0%

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

   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. *-un-lft-identity 99.7%

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

   2. associate-/l* 99.7%

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

   3. frac-times 99.7%

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

   4. *-un-lft-identity 99.7%

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

   5. +-commutative 99.7%

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

   6. fma-def 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{r \cdot w}\\ t_1 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{elif}\;v \leq 1.6 \cdot 10^{-63}:\\ \;\;\;\;-4.5 + \left(t_1 - \frac{0.125 \cdot \left(3 + v \cdot -2\right)}{t_0 \cdot t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(t_1 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* r w))) (t_1 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= v -2.5e-13)
     (+ (/ (/ 2.0 r) r) (- -1.5 (/ r (/ 4.0 (* w (* r w))))))
     (if (<= v 1.6e-63)
       (+ -4.5 (- t_1 (/ (* 0.125 (+ 3.0 (* v -2.0))) (* t_0 t_0))))
       (+ -4.5 (- t_1 (* (* (* r w) (* r w)) 0.25)))))))

C

double code(double v, double w, double r) {
	double t_0 = 1.0 / (r * w);
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= -2.5e-13) {
		tmp = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	} else if (v <= 1.6e-63) {
		tmp = -4.5 + (t_1 - ((0.125 * (3.0 + (v * -2.0))) / (t_0 * t_0)));
	} else {
		tmp = -4.5 + (t_1 - (((r * w) * (r * w)) * 0.25));
	}
	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 = 1.0d0 / (r * w)
    t_1 = 3.0d0 + (2.0d0 / (r * r))
    if (v <= (-2.5d-13)) then
        tmp = ((2.0d0 / r) / r) + ((-1.5d0) - (r / (4.0d0 / (w * (r * w)))))
    else if (v <= 1.6d-63) then
        tmp = (-4.5d0) + (t_1 - ((0.125d0 * (3.0d0 + (v * (-2.0d0)))) / (t_0 * t_0)))
    else
        tmp = (-4.5d0) + (t_1 - (((r * w) * (r * w)) * 0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 1.0 / (r * w);
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= -2.5e-13) {
		tmp = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	} else if (v <= 1.6e-63) {
		tmp = -4.5 + (t_1 - ((0.125 * (3.0 + (v * -2.0))) / (t_0 * t_0)));
	} else {
		tmp = -4.5 + (t_1 - (((r * w) * (r * w)) * 0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 1.0 / (r * w)
	t_1 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if v <= -2.5e-13:
		tmp = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))))
	elif v <= 1.6e-63:
		tmp = -4.5 + (t_1 - ((0.125 * (3.0 + (v * -2.0))) / (t_0 * t_0)))
	else:
		tmp = -4.5 + (t_1 - (((r * w) * (r * w)) * 0.25))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(1.0 / Float64(r * w))
	t_1 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (v <= -2.5e-13)
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(r / Float64(4.0 / Float64(w * Float64(r * w))))));
	elseif (v <= 1.6e-63)
		tmp = Float64(-4.5 + Float64(t_1 - Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) / Float64(t_0 * t_0))));
	else
		tmp = Float64(-4.5 + Float64(t_1 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 1.0 / (r * w);
	t_1 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if (v <= -2.5e-13)
		tmp = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	elseif (v <= 1.6e-63)
		tmp = -4.5 + (t_1 - ((0.125 * (3.0 + (v * -2.0))) / (t_0 * t_0)));
	else
		tmp = -4.5 + (t_1 - (((r * w) * (r * w)) * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2.5e-13], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(r / N[(4.0 / N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1.6e-63], N[(-4.5 + N[(t$95$1 - N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(t$95$1 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.49999999999999995e-13

   1. Initial program 85.7%

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

      1. associate--l- 85.7%

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

      2. +-commutative 85.7%

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

      3. associate--l+ 85.7%

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

      4. +-commutative 85.7%

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

      5. associate--r+ 85.7%

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

      6. metadata-eval 85.7%

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

      7. associate-*r* 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* 87.0%

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

      10. *-commutative 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in v around inf 94.0%

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

      1. unpow2 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

   10. Simplified 99.7%

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

       1. inv-pow 99.7%

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

       2. associate-/r/ 99.7%

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

       3. unpow-prod-down 99.7%

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

       4. inv-pow 99.7%

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

       5. clear-num 99.7%

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

       6. inv-pow 99.7%

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

   12. Applied egg-rr 99.7%

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

       1. un-div-inv 99.8%

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

   14. Applied egg-rr 99.8%

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

   if -2.49999999999999995e-13 < v < 1.59999999999999994e-63

   1. Initial program 84.7%

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

   3. Simplified 81.7%

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

      1. *-un-lft-identity 81.7%

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

      2. add-sqr-sqrt 81.6%

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

      3. times-frac 81.6%

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

      4. unswap-sqr 81.7%

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

      5. sqrt-prod 52.9%

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

      6. add-sqr-sqrt 62.2%

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

      7. unswap-sqr 78.6%

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

      8. sqrt-prod 62.7%

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

      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. Taylor expanded in v around 0 99.7%

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

   if 1.59999999999999994e-63 < v

   1. Initial program 82.7%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in v around inf 81.2%

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

      1. *-commutative 81.2%

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

      2. unpow2 81.2%

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

      3. unpow2 81.2%

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

      4. *-commutative 81.2%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{elif}\;v \leq 1.6 \cdot 10^{-63}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + v \cdot -2\right)}{\frac{1}{r \cdot w} \cdot \frac{1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \]

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 84.3%

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

3. Simplified 83.3%

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

   1. *-un-lft-identity 83.3%

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

   2. add-sqr-sqrt 83.2%

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

   3. times-frac 83.2%

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

   4. unswap-sqr 83.2%

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

   5. sqrt-prod 50.4%

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

   6. add-sqr-sqrt 64.1%

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

   7. unswap-sqr 78.8%

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

   8. sqrt-prod 58.0%

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

   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: 95.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -4.5 \cdot 10^{+69} \lor \neg \left(v \leq 5 \cdot 10^{-66}\right) \land v \leq 1.2 \cdot 10^{+187}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -4.5e+69) (and (not (<= v 5e-66)) (<= v 1.2e+187)))
     (+ t_0 (- -1.5 (* (/ r 4.0) (* w (* 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 <= -4.5e+69) || (!(v <= 5e-66) && (v <= 1.2e+187))) {
		tmp = t_0 + (-1.5 - ((r / 4.0) * (w * (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 <= (-4.5d+69)) .or. (.not. (v <= 5d-66)) .and. (v <= 1.2d+187)) then
        tmp = t_0 + ((-1.5d0) - ((r / 4.0d0) * (w * (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 <= -4.5e+69) || (!(v <= 5e-66) && (v <= 1.2e+187))) {
		tmp = t_0 + (-1.5 - ((r / 4.0) * (w * (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 <= -4.5e+69) or (not (v <= 5e-66) and (v <= 1.2e+187)):
		tmp = t_0 + (-1.5 - ((r / 4.0) * (w * (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 <= -4.5e+69) || (!(v <= 5e-66) && (v <= 1.2e+187)))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r / 4.0) * Float64(w * Float64(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 <= -4.5e+69) || (~((v <= 5e-66)) && (v <= 1.2e+187)))
		tmp = t_0 + (-1.5 - ((r / 4.0) * (w * (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, -4.5e+69], And[N[Not[LessEqual[v, 5e-66]], $MachinePrecision], LessEqual[v, 1.2e+187]]], N[(t$95$0 + N[(-1.5 - N[(N[(r / 4.0), $MachinePrecision] * N[(w * N[(r * w), $MachinePrecision]), $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 < -4.4999999999999999e69 or 4.99999999999999962e-66 < v < 1.19999999999999993e187

   1. Initial program 82.1%

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

      1. associate--l- 82.1%

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

      2. +-commutative 82.1%

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

      3. associate--l+ 82.1%

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

      4. +-commutative 82.1%

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

      5. associate--r+ 82.1%

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

      6. metadata-eval 82.1%

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

      7. associate-*r* 81.1%

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

      8. *-commutative 81.1%

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

      9. associate-/l* 84.0%

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

      10. *-commutative 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in v around inf 88.9%

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

      1. unpow2 88.9%

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

      2. *-commutative 88.9%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. associate-/r/ 99.8%

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

      2. *-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   if -4.4999999999999999e69 < v < 4.99999999999999962e-66 or 1.19999999999999993e187 < v

   1. Initial program 85.7%

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

      1. associate--l- 85.7%

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

      2. +-commutative 85.7%

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

      3. associate--l+ 85.7%

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

      4. +-commutative 85.7%

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

      5. associate--r+ 85.7%

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

      6. metadata-eval 85.7%

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

      7. associate-*r* 85.7%

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

      8. *-commutative 85.7%

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

      9. associate-/l* 85.6%

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

      10. *-commutative 85.6%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in v around 0 85.6%

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

      1. unpow2 85.6%

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

      2. *-commutative 85.6%

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

      3. associate-*l* 96.0%

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

      4. *-commutative 96.0%

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

   6. Simplified 96.0%

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

      1. associate-/r/ 96.0%

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

      2. *-commutative 96.0%

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

      3. associate-*r* 99.7%

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

      4. div-inv 99.8%

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

      5. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4.5 \cdot 10^{+69} \lor \neg \left(v \leq 5 \cdot 10^{-66}\right) \land v \leq 1.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -8 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{elif}\;v \leq 1.65 \cdot 10^{-63}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -8e-14)
     (+ (/ (/ 2.0 r) r) (- -1.5 (/ r (/ 4.0 (* w (* r w))))))
     (if (<= v 1.65e-63)
       (+ t_0 (- -1.5 (* (* r w) (* w (* r 0.375)))))
       (+ -4.5 (- (+ 3.0 t_0) (* (* (* r w) (* r w)) 0.25)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -8e-14) {
		tmp = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	} else if (v <= 1.65e-63) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (((r * w) * (r * w)) * 0.25));
	}
	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 <= (-8d-14)) then
        tmp = ((2.0d0 / r) / r) + ((-1.5d0) - (r / (4.0d0 / (w * (r * w)))))
    else if (v <= 1.65d-63) then
        tmp = t_0 + ((-1.5d0) - ((r * w) * (w * (r * 0.375d0))))
    else
        tmp = (-4.5d0) + ((3.0d0 + t_0) - (((r * w) * (r * w)) * 0.25d0))
    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 <= -8e-14) {
		tmp = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	} else if (v <= 1.65e-63) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (((r * w) * (r * w)) * 0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -8e-14:
		tmp = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))))
	elif v <= 1.65e-63:
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))))
	else:
		tmp = -4.5 + ((3.0 + t_0) - (((r * w) * (r * w)) * 0.25))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -8e-14)
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(r / Float64(4.0 / Float64(w * Float64(r * w))))));
	elseif (v <= 1.65e-63)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))));
	else
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -8e-14)
		tmp = ((2.0 / r) / r) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	elseif (v <= 1.65e-63)
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.375))));
	else
		tmp = -4.5 + ((3.0 + t_0) - (((r * w) * (r * w)) * 0.25));
	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, -8e-14], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(r / N[(4.0 / N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1.65e-63], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -7.99999999999999999e-14

   1. Initial program 85.7%

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

      1. associate--l- 85.7%

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

      2. +-commutative 85.7%

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

      3. associate--l+ 85.7%

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

      4. +-commutative 85.7%

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

      5. associate--r+ 85.7%

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

      6. metadata-eval 85.7%

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

      7. associate-*r* 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* 87.0%

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

      10. *-commutative 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in v around inf 94.0%

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

      1. unpow2 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/l* 99.7%

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

   10. Simplified 99.7%

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

       1. inv-pow 99.7%

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

       2. associate-/r/ 99.7%

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

       3. unpow-prod-down 99.7%

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

       4. inv-pow 99.7%

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

       5. clear-num 99.7%

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

       6. inv-pow 99.7%

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

   12. Applied egg-rr 99.7%

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

       1. un-div-inv 99.8%

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

   14. Applied egg-rr 99.8%

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

   if -7.99999999999999999e-14 < v < 1.64999999999999997e-63

   1. Initial program 84.7%

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

      1. associate--l- 84.7%

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

      2. +-commutative 84.7%

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

      3. associate--l+ 84.7%

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

      4. +-commutative 84.7%

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

      5. associate--r+ 84.7%

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

      6. metadata-eval 84.7%

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

      7. associate-*r* 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-/l* 84.6%

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

      10. *-commutative 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in v around 0 84.6%

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

      1. unpow2 84.6%

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

      2. *-commutative 84.6%

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

      3. associate-*l* 95.2%

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

      4. *-commutative 95.2%

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

   6. Simplified 95.2%

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

      1. associate-/r/ 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*r* 99.7%

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

      4. div-inv 99.7%

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

      5. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 1.64999999999999997e-63 < v

   1. Initial program 82.7%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in v around inf 81.2%

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

      1. *-commutative 81.2%

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

      2. unpow2 81.2%

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

      3. unpow2 81.2%

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

      4. *-commutative 81.2%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -8 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{elif}\;v \leq 1.65 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \]

Alternative 7: 93.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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) - ((r * w) * (w * (r * 0.375d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 83.9%

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

   8. *-commutative 83.9%

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

   9. associate-/l* 85.0%

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

   10. *-commutative 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in v around 0 83.2%

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

   1. unpow2 83.2%

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

   2. *-commutative 83.2%

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

   3. associate-*l* 92.6%

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

   4. *-commutative 92.6%

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

6. Simplified 92.6%

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

   1. associate-/r/ 92.6%

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

   2. *-commutative 92.6%

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

   3. associate-*r* 94.9%

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

   4. div-inv 94.9%

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

   5. metadata-eval 94.9%

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

8. Applied egg-rr 94.9%

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

9. Final simplification 94.9%

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

Alternative 8: 56.6% 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 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* 83.9%

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

   8. *-commutative 83.9%

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

   9. associate-/l* 85.0%

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

   10. *-commutative 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in v around 0 83.2%

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

   1. unpow2 83.2%

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

   2. *-commutative 83.2%

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

   3. associate-*l* 92.6%

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

   4. *-commutative 92.6%

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

6. Simplified 92.6%

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

   1. associate-/r/ 92.6%

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

   2. *-commutative 92.6%

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

   3. associate-*r* 94.9%

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

   4. div-inv 94.9%

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

   5. metadata-eval 94.9%

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

8. Applied egg-rr 94.9%

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

9. Taylor expanded in r around 0 56.6%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
10. Step-by-step derivation

    1. sub-neg 56.6%

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

    2. metadata-eval 56.6%

       \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{-1.5} \]

    3. unpow2 56.6%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

    4. associate-*r/ 56.6%

       \[\leadsto \color{blue}{\frac{2 \cdot 1}{r \cdot r}} + -1.5 \]

    5. metadata-eval 56.6%

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

11. Simplified 56.6%

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

12. Final simplification 56.6%

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

Alternative 9: 43.7% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 83.9%

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

   8. *-commutative 83.9%

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

   9. associate-/l* 85.0%

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

   10. *-commutative 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in v around 0 83.2%

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

   1. unpow2 83.2%

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

   2. *-commutative 83.2%

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

   3. associate-*l* 92.6%

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

   4. *-commutative 92.6%

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

6. Simplified 92.6%

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

   1. associate-/r/ 92.6%

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

   2. *-commutative 92.6%

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

   3. associate-*r* 94.9%

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

   4. div-inv 94.9%

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

   5. metadata-eval 94.9%

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

8. Applied egg-rr 94.9%

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

9. Taylor expanded in r around 0 47.8%

   \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
10. Step-by-step derivation

    1. unpow2 47.8%

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

11. Simplified 47.8%

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

12. Final simplification 47.8%

    \[\leadsto \frac{2}{r \cdot r} \]

Reproduce

herbie shell --seed 2023279 
(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))