Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 99.7%

Time: 13.0s

Alternatives: 13

Speedup: 1.0×

• 53.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. associate--l- 86.3%

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

   2. associate-*l* 82.0%

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

   3. sqr-neg 82.0%

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

   4. associate-*l* 86.3%

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

   5. associate-/l* 89.1%

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

   6. fma-define 89.1%

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

3. Simplified 89.1%

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

   1. add-sqr-sqrt 89.1%

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

   2. *-un-lft-identity 89.1%

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

   3. times-frac 89.1%

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

   4. associate-*r* 83.7%

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

   5. sqrt-prod 83.7%

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

   6. sqrt-prod 45.5%

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

   7. add-sqr-sqrt 70.0%

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

   8. sqrt-prod 30.0%

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

   9. add-sqr-sqrt 68.5%

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

   10. associate-*r* 62.7%

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

   11. sqrt-prod 62.7%

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

   12. sqrt-prod 36.1%

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

   13. add-sqr-sqrt 71.8%

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

   14. sqrt-prod 44.8%

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

   15. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{-145}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 540000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\frac{r}{1 - v} \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v - 1}\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.5e-145)
   (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5)
   (if (<= r 540000000000.0)
     (+ (/ 2.0 (* r r)) (- -1.5 (* 0.375 (* r (* (/ r (- 1.0 v)) (* w w))))))
     (+
      3.0
      (-
       (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (* w (/ r (- v 1.0)))))
       4.5)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.5e-145) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 540000000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (v - 1.0))))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5.5d-145) then
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - 4.5d0
    else if (r <= 540000000000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (0.375d0 * (r * ((r / (1.0d0 - v)) * (w * w)))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (w * (r / (v - 1.0d0))))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.5e-145) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 540000000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (v - 1.0))))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.5e-145:
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5
	elif r <= 540000000000.0:
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (v - 1.0))))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.5e-145)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - 4.5);
	elseif (r <= 540000000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(Float64(r / Float64(1.0 - v)) * Float64(w * w))))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(w * Float64(r / Float64(v - 1.0))))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.5e-145)
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	elseif (r <= 540000000000.0)
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (v - 1.0))))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.5e-145], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 540000000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(0.375 * N[(r * N[(N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / N[(v - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 5.50000000000000015e-145

   1. Initial program 83.5%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in r around 0 62.9%

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

      1. associate-/r* 62.9%

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

      2. div-inv 62.9%

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

   7. Applied egg-rr 62.9%

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

      1. associate-*r/ 62.9%

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

      2. *-rgt-identity 62.9%

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

   9. Simplified 62.9%

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

   if 5.50000000000000015e-145 < r < 5.4e11

   1. Initial program 90.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in v around 0 87.1%

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

   if 5.4e11 < r

   1. Initial program 91.4%

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

      1. associate--l- 91.4%

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

      2. associate-*l* 82.3%

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

      3. sqr-neg 82.3%

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

      4. associate-*l* 91.4%

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

      5. associate-/l* 96.0%

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

      6. fma-define 96.0%

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

   3. Simplified 96.0%

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

      1. associate-/l* 96.0%

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

      2. *-commutative 96.0%

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

      3. associate-*r/ 95.9%

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

      4. associate-*l* 97.4%

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

      5. associate-*r* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in r around inf 99.8%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{-145}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 540000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\frac{r}{1 - v} \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v - 1}\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \frac{r}{v - 1}\\ \mathbf{if}\;r \leq 540000000000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(w \cdot \left(r \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right) \cdot t\_0\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot t\_0\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (/ r (- v 1.0)))))
   (if (<= r 540000000000.0)
     (-
      (+ (+ 3.0 (/ 2.0 (* r r))) (* (* w (* r (+ (* v -0.25) 0.375))) t_0))
      4.5)
     (+ 3.0 (- (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) t_0)) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = w * (r / (v - 1.0));
	double tmp;
	if (r <= 540000000000.0) {
		tmp = ((3.0 + (2.0 / (r * r))) + ((w * (r * ((v * -0.25) + 0.375))) * t_0)) - 4.5;
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * t_0)) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w * (r / (v - 1.0d0))
    if (r <= 540000000000.0d0) then
        tmp = ((3.0d0 + (2.0d0 / (r * r))) + ((w * (r * ((v * (-0.25d0)) + 0.375d0))) * t_0)) - 4.5d0
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * t_0)) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = w * (r / (v - 1.0));
	double tmp;
	if (r <= 540000000000.0) {
		tmp = ((3.0 + (2.0 / (r * r))) + ((w * (r * ((v * -0.25) + 0.375))) * t_0)) - 4.5;
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * t_0)) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = w * (r / (v - 1.0))
	tmp = 0
	if r <= 540000000000.0:
		tmp = ((3.0 + (2.0 / (r * r))) + ((w * (r * ((v * -0.25) + 0.375))) * t_0)) - 4.5
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * t_0)) - 4.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(r / Float64(v - 1.0)))
	tmp = 0.0
	if (r <= 540000000000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(Float64(w * Float64(r * Float64(Float64(v * -0.25) + 0.375))) * t_0)) - 4.5);
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * t_0)) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = w * (r / (v - 1.0));
	tmp = 0.0;
	if (r <= 540000000000.0)
		tmp = ((3.0 + (2.0 / (r * r))) + ((w * (r * ((v * -0.25) + 0.375))) * t_0)) - 4.5;
	else
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * t_0)) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(r / N[(v - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 540000000000.0], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(w * N[(r * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 5.4e11

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* 87.0%

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

      2. cancel-sign-sub-inv 87.0%

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

      3. metadata-eval 87.0%

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

      4. +-commutative 87.0%

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

      5. *-commutative 87.0%

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

      6. fma-undefine 87.0%

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

      7. *-commutative 87.0%

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

      8. *-commutative 87.0%

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

      9. associate-/l* 87.0%

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

      10. *-commutative 87.0%

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

      11. associate-*r/ 87.0%

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

      12. associate-*r* 83.6%

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

      13. associate-*l* 92.7%

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

      14. associate-*r* 93.6%

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

   4. Applied egg-rr 93.6%

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

   if 5.4e11 < r

   1. Initial program 91.4%

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

      1. associate--l- 91.4%

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

      2. associate-*l* 82.3%

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

      3. sqr-neg 82.3%

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

      4. associate-*l* 91.4%

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

      5. associate-/l* 96.0%

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

      6. fma-define 96.0%

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

   3. Simplified 96.0%

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

      1. associate-/l* 96.0%

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

      2. *-commutative 96.0%

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

      3. associate-*r/ 95.9%

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

      4. associate-*l* 97.4%

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

      5. associate-*r* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in r around inf 99.8%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 540000000000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(w \cdot \left(r \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v - 1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v - 1}\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{-145}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 550000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\frac{r}{1 - v} \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{r \cdot \left(w \cdot \left(v \cdot -0.25 + 0.375\right)\right)}{\frac{v - 1}{r \cdot w}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.5e-145)
   (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5)
   (if (<= r 550000000000.0)
     (+ (/ 2.0 (* r r)) (- -1.5 (* 0.375 (* r (* (/ r (- 1.0 v)) (* w w))))))
     (+
      3.0
      (- (/ (* r (* w (+ (* v -0.25) 0.375))) (/ (- v 1.0) (* r w))) 4.5)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.5e-145) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 550000000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))));
	} else {
		tmp = 3.0 + (((r * (w * ((v * -0.25) + 0.375))) / ((v - 1.0) / (r * w))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5.5d-145) then
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - 4.5d0
    else if (r <= 550000000000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (0.375d0 * (r * ((r / (1.0d0 - v)) * (w * w)))))
    else
        tmp = 3.0d0 + (((r * (w * ((v * (-0.25d0)) + 0.375d0))) / ((v - 1.0d0) / (r * w))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.5e-145) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 550000000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))));
	} else {
		tmp = 3.0 + (((r * (w * ((v * -0.25) + 0.375))) / ((v - 1.0) / (r * w))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.5e-145:
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5
	elif r <= 550000000000.0:
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))))
	else:
		tmp = 3.0 + (((r * (w * ((v * -0.25) + 0.375))) / ((v - 1.0) / (r * w))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.5e-145)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - 4.5);
	elseif (r <= 550000000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(Float64(r / Float64(1.0 - v)) * Float64(w * w))))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(r * Float64(w * Float64(Float64(v * -0.25) + 0.375))) / Float64(Float64(v - 1.0) / Float64(r * w))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.5e-145)
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	elseif (r <= 550000000000.0)
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))));
	else
		tmp = 3.0 + (((r * (w * ((v * -0.25) + 0.375))) / ((v - 1.0) / (r * w))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.5e-145], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 550000000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(0.375 * N[(r * N[(N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(r * N[(w * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v - 1.0), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 5.50000000000000015e-145

   1. Initial program 83.5%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in r around 0 62.9%

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

      1. associate-/r* 62.9%

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

      2. div-inv 62.9%

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

   7. Applied egg-rr 62.9%

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

      1. associate-*r/ 62.9%

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

      2. *-rgt-identity 62.9%

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

   9. Simplified 62.9%

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

   if 5.50000000000000015e-145 < r < 5.5e11

   1. Initial program 90.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in v around 0 87.1%

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

   if 5.5e11 < r

   1. Initial program 91.4%

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

      1. associate--l- 91.4%

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

      2. associate-*l* 82.3%

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

      3. sqr-neg 82.3%

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

      4. associate-*l* 91.4%

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

      5. associate-/l* 96.0%

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

      6. fma-define 96.0%

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

   3. Simplified 96.0%

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

      1. add-sqr-sqrt 95.9%

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

      2. *-un-lft-identity 95.9%

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

      3. times-frac 95.9%

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

      4. associate-*r* 85.3%

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

      5. sqrt-prod 85.3%

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

      6. sqrt-prod 95.9%

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

      7. add-sqr-sqrt 95.9%

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

      8. sqrt-prod 45.0%

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

      9. add-sqr-sqrt 55.1%

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

      10. associate-*r* 47.6%

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

      11. sqrt-prod 47.6%

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

      12. sqrt-prod 55.1%

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

      13. add-sqr-sqrt 55.1%

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

      14. sqrt-prod 45.8%

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

      15. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-*r* 98.2%

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

      3. clear-num 98.2%

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

      4. un-div-inv 98.2%

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

      5. distribute-lft-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. associate-*r* 98.2%

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

      8. metadata-eval 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in r around 0 98.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{-145}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 550000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\frac{r}{1 - v} \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{r \cdot \left(w \cdot \left(v \cdot -0.25 + 0.375\right)\right)}{\frac{v - 1}{r \cdot w}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{-145}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 1.6 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\frac{r}{1 - v} \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{\left(r \cdot w\right) \cdot \left(v \cdot -0.25 + 0.375\right)}{\frac{\frac{v}{r}}{w}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.5e-145)
   (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5)
   (if (<= r 1.6e+144)
     (+ (/ 2.0 (* r r)) (- -1.5 (* 0.375 (* r (* (/ r (- 1.0 v)) (* w w))))))
     (+ 3.0 (- (/ (* (* r w) (+ (* v -0.25) 0.375)) (/ (/ v r) w)) 4.5)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.5e-145) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 1.6e+144) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))));
	} else {
		tmp = 3.0 + ((((r * w) * ((v * -0.25) + 0.375)) / ((v / r) / w)) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5.5d-145) then
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - 4.5d0
    else if (r <= 1.6d+144) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (0.375d0 * (r * ((r / (1.0d0 - v)) * (w * w)))))
    else
        tmp = 3.0d0 + ((((r * w) * ((v * (-0.25d0)) + 0.375d0)) / ((v / r) / w)) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.5e-145) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 1.6e+144) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))));
	} else {
		tmp = 3.0 + ((((r * w) * ((v * -0.25) + 0.375)) / ((v / r) / w)) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.5e-145:
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5
	elif r <= 1.6e+144:
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))))
	else:
		tmp = 3.0 + ((((r * w) * ((v * -0.25) + 0.375)) / ((v / r) / w)) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.5e-145)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - 4.5);
	elseif (r <= 1.6e+144)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(Float64(r / Float64(1.0 - v)) * Float64(w * w))))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(Float64(r * w) * Float64(Float64(v * -0.25) + 0.375)) / Float64(Float64(v / r) / w)) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.5e-145)
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	elseif (r <= 1.6e+144)
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((r / (1.0 - v)) * (w * w)))));
	else
		tmp = 3.0 + ((((r * w) * ((v * -0.25) + 0.375)) / ((v / r) / w)) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.5e-145], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 1.6e+144], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(0.375 * N[(r * N[(N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(v / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 5.50000000000000015e-145

   1. Initial program 83.5%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in r around 0 62.9%

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

      1. associate-/r* 62.9%

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

      2. div-inv 62.9%

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

   7. Applied egg-rr 62.9%

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

      1. associate-*r/ 62.9%

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

      2. *-rgt-identity 62.9%

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

   9. Simplified 62.9%

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

   if 5.50000000000000015e-145 < r < 1.6e144

   1. Initial program 92.3%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in v around 0 76.6%

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

   if 1.6e144 < r

   1. Initial program 88.9%

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

      1. associate--l- 88.9%

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

      2. associate-*l* 70.5%

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

      3. sqr-neg 70.5%

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

      4. associate-*l* 88.9%

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

      5. associate-/l* 92.0%

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

      6. fma-define 92.0%

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

   3. Simplified 92.0%

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

      1. add-sqr-sqrt 92.0%

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

      2. *-un-lft-identity 92.0%

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

      3. times-frac 92.0%

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

      4. associate-*r* 70.5%

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

      5. sqrt-prod 70.5%

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

      6. sqrt-prod 92.0%

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

      7. add-sqr-sqrt 92.0%

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

      8. sqrt-prod 41.6%

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

      9. add-sqr-sqrt 48.8%

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

      10. associate-*r* 33.6%

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

      11. sqrt-prod 33.6%

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

      12. sqrt-prod 48.7%

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

      13. add-sqr-sqrt 48.8%

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

      14. sqrt-prod 43.3%

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

      15. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in r around inf 99.8%

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

      1. /-rgt-identity 99.8%

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

      2. associate-*r* 99.7%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r* 99.8%

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

      8. metadata-eval 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in v around inf 80.2%

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

       1. mul-1-neg 80.2%

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

       2. associate-/r* 80.2%

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

   12. Simplified 80.2%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{-145}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 1.6 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\frac{r}{1 - v} \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{\left(r \cdot w\right) \cdot \left(v \cdot -0.25 + 0.375\right)}{\frac{\frac{v}{r}}{w}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 112:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{elif}\;r \leq 1.26 \cdot 10^{+144}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{1 - v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{\left(r \cdot w\right) \cdot \left(v \cdot -0.25 + 0.375\right)}{\frac{\frac{v}{r}}{w}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 112.0)
   (- (+ 3.0 (/ 2.0 (* r r))) 4.5)
   (if (<= r 1.26e+144)
     (- 3.0 (+ 4.5 (/ (* (* r w) 0.375) (/ (- 1.0 v) (* r w)))))
     (+ 3.0 (- (/ (* (* r w) (+ (* v -0.25) 0.375)) (/ (/ v r) w)) 4.5)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 112.0) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else if (r <= 1.26e+144) {
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))));
	} else {
		tmp = 3.0 + ((((r * w) * ((v * -0.25) + 0.375)) / ((v / r) / w)) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 112.0d0) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - 4.5d0
    else if (r <= 1.26d+144) then
        tmp = 3.0d0 - (4.5d0 + (((r * w) * 0.375d0) / ((1.0d0 - v) / (r * w))))
    else
        tmp = 3.0d0 + ((((r * w) * ((v * (-0.25d0)) + 0.375d0)) / ((v / r) / w)) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 112.0) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else if (r <= 1.26e+144) {
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))));
	} else {
		tmp = 3.0 + ((((r * w) * ((v * -0.25) + 0.375)) / ((v / r) / w)) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 112.0:
		tmp = (3.0 + (2.0 / (r * r))) - 4.5
	elif r <= 1.26e+144:
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))))
	else:
		tmp = 3.0 + ((((r * w) * ((v * -0.25) + 0.375)) / ((v / r) / w)) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 112.0)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - 4.5);
	elseif (r <= 1.26e+144)
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * 0.375) / Float64(Float64(1.0 - v) / Float64(r * w)))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(Float64(r * w) * Float64(Float64(v * -0.25) + 0.375)) / Float64(Float64(v / r) / w)) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 112.0)
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	elseif (r <= 1.26e+144)
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))));
	else
		tmp = 3.0 + ((((r * w) * ((v * -0.25) + 0.375)) / ((v / r) / w)) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 112.0], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 1.26e+144], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(v / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 112

   1. Initial program 84.6%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in r around 0 64.0%

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

   if 112 < r < 1.26000000000000001e144

   1. Initial program 93.9%

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

      1. associate--l- 93.9%

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

      2. associate-*l* 94.0%

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

      3. sqr-neg 94.0%

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

      4. associate-*l* 93.9%

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

      5. associate-/l* 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. times-frac 99.5%

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

      4. associate-*r* 99.6%

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

      5. sqrt-prod 99.6%

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

      6. sqrt-prod 99.6%

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

      7. add-sqr-sqrt 99.6%

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

      8. sqrt-prod 49.8%

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

      9. add-sqr-sqrt 62.4%

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

      10. associate-*r* 62.4%

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

      11. sqrt-prod 62.4%

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

      12. sqrt-prod 62.4%

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

      13. add-sqr-sqrt 62.4%

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

      14. sqrt-prod 49.8%

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

      15. add-sqr-sqrt 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in r around inf 99.5%

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

      1. /-rgt-identity 99.5%

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

      2. associate-*r* 96.9%

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

      3. clear-num 96.8%

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

      4. un-div-inv 96.8%

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

      5. distribute-lft-in 96.8%

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

      6. metadata-eval 96.8%

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

      7. associate-*r* 96.8%

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

      8. metadata-eval 96.8%

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

   9. Applied egg-rr 96.8%

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

   10. Taylor expanded in v around 0 67.1%

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

   if 1.26000000000000001e144 < r

   1. Initial program 88.9%

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

      1. associate--l- 88.9%

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

      2. associate-*l* 70.5%

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

      3. sqr-neg 70.5%

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

      4. associate-*l* 88.9%

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

      5. associate-/l* 92.0%

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

      6. fma-define 92.0%

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

   3. Simplified 92.0%

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

      1. add-sqr-sqrt 92.0%

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

      2. *-un-lft-identity 92.0%

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

      3. times-frac 92.0%

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

      4. associate-*r* 70.5%

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

      5. sqrt-prod 70.5%

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

      6. sqrt-prod 92.0%

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

      7. add-sqr-sqrt 92.0%

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

      8. sqrt-prod 41.6%

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

      9. add-sqr-sqrt 48.8%

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

      10. associate-*r* 33.6%

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

      11. sqrt-prod 33.6%

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

      12. sqrt-prod 48.7%

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

      13. add-sqr-sqrt 48.8%

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

      14. sqrt-prod 43.3%

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

      15. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in r around inf 99.8%

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

      1. /-rgt-identity 99.8%

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

      2. associate-*r* 99.7%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r* 99.8%

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

      8. metadata-eval 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in v around inf 80.2%

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

       1. mul-1-neg 80.2%

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

       2. associate-/r* 80.2%

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

   12. Simplified 80.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 112:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{elif}\;r \leq 1.26 \cdot 10^{+144}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{1 - v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{\left(r \cdot w\right) \cdot \left(v \cdot -0.25 + 0.375\right)}{\frac{\frac{v}{r}}{w}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 7500:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{elif}\;r \leq 4.1 \cdot 10^{+93}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{1 - v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)}{\frac{v - 1}{r \cdot w}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 7500.0)
   (- (+ 3.0 (/ 2.0 (* r r))) 4.5)
   (if (<= r 4.1e+93)
     (- 3.0 (+ 4.5 (/ (* (* r w) 0.375) (/ (- 1.0 v) (* r w)))))
     (+ 3.0 (- (/ (* -0.25 (* r (* v w))) (/ (- v 1.0) (* r w))) 4.5)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 7500.0) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else if (r <= 4.1e+93) {
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))));
	} else {
		tmp = 3.0 + (((-0.25 * (r * (v * w))) / ((v - 1.0) / (r * w))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 7500.0d0) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - 4.5d0
    else if (r <= 4.1d+93) then
        tmp = 3.0d0 - (4.5d0 + (((r * w) * 0.375d0) / ((1.0d0 - v) / (r * w))))
    else
        tmp = 3.0d0 + ((((-0.25d0) * (r * (v * w))) / ((v - 1.0d0) / (r * w))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 7500.0) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else if (r <= 4.1e+93) {
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))));
	} else {
		tmp = 3.0 + (((-0.25 * (r * (v * w))) / ((v - 1.0) / (r * w))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 7500.0:
		tmp = (3.0 + (2.0 / (r * r))) - 4.5
	elif r <= 4.1e+93:
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))))
	else:
		tmp = 3.0 + (((-0.25 * (r * (v * w))) / ((v - 1.0) / (r * w))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 7500.0)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - 4.5);
	elseif (r <= 4.1e+93)
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * 0.375) / Float64(Float64(1.0 - v) / Float64(r * w)))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(-0.25 * Float64(r * Float64(v * w))) / Float64(Float64(v - 1.0) / Float64(r * w))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 7500.0)
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	elseif (r <= 4.1e+93)
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))));
	else
		tmp = 3.0 + (((-0.25 * (r * (v * w))) / ((v - 1.0) / (r * w))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 7500.0], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 4.1e+93], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(-0.25 * N[(r * N[(v * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v - 1.0), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 7500

   1. Initial program 84.6%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in r around 0 64.0%

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

   if 7500 < r < 4.1000000000000001e93

   1. Initial program 94.7%

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

      1. associate--l- 94.7%

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

      2. associate-*l* 94.7%

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

      3. sqr-neg 94.7%

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

      4. associate-*l* 94.7%

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

      5. associate-/l* 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. times-frac 99.6%

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

      4. associate-*r* 99.6%

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

      5. sqrt-prod 99.7%

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

      6. sqrt-prod 99.7%

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

      7. add-sqr-sqrt 99.7%

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

      8. sqrt-prod 55.3%

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

      9. add-sqr-sqrt 66.6%

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

      10. associate-*r* 66.6%

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

      11. sqrt-prod 66.6%

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

      12. sqrt-prod 66.6%

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

      13. add-sqr-sqrt 66.6%

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

      14. sqrt-prod 55.3%

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

      15. add-sqr-sqrt 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in r around inf 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-*r* 99.7%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r* 99.8%

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

      8. metadata-eval 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in v around 0 78.8%

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

   if 4.1000000000000001e93 < r

   1. Initial program 90.2%

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

      1. associate--l- 90.2%

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

      2. associate-*l* 77.7%

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

      3. sqr-neg 77.7%

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

      4. associate-*l* 90.2%

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

      5. associate-/l* 94.4%

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

      6. fma-define 94.4%

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

   3. Simplified 94.4%

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

      1. add-sqr-sqrt 94.4%

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

      2. *-un-lft-identity 94.4%

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

      3. times-frac 94.4%

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

      4. associate-*r* 79.7%

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

      5. sqrt-prod 79.7%

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

      6. sqrt-prod 94.4%

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

      7. add-sqr-sqrt 94.4%

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

      8. sqrt-prod 42.0%

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

      9. add-sqr-sqrt 51.4%

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

      10. associate-*r* 41.0%

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

      11. sqrt-prod 41.0%

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

      12. sqrt-prod 51.4%

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

      13. add-sqr-sqrt 51.4%

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

      14. sqrt-prod 43.1%

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

      15. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-*r* 97.7%

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

      3. clear-num 97.7%

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

      4. un-div-inv 97.6%

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

      5. distribute-lft-in 97.6%

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

      6. metadata-eval 97.6%

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

      7. associate-*r* 97.6%

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

      8. metadata-eval 97.6%

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

   9. Applied egg-rr 97.6%

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

   10. Taylor expanded in v around inf 75.3%

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

       1. *-commutative 75.3%

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

       2. *-commutative 75.3%

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

   12. Simplified 75.3%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7500:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{elif}\;r \leq 4.1 \cdot 10^{+93}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{1 - v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)}{\frac{v - 1}{r \cdot w}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. associate--l- 86.3%

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

   2. associate-*l* 82.0%

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

   3. sqr-neg 82.0%

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

   4. associate-*l* 86.3%

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

   5. associate-/l* 89.1%

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

   6. fma-define 89.1%

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

3. Simplified 89.1%

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

   1. associate-/l* 89.1%

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

   2. *-commutative 89.1%

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

   3. associate-*r/ 89.2%

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

   4. associate-*l* 97.5%

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

   5. associate-*r* 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 9: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 540000000000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(r \cdot w\right) \cdot 0.375\right) \cdot \left(r \cdot \frac{w}{v - 1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v - 1}\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 540000000000.0)
   (-
    (+ (+ 3.0 (/ 2.0 (* r r))) (* (* (* r w) 0.375) (* r (/ w (- v 1.0)))))
    4.5)
   (+
    3.0
    (-
     (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (* w (/ r (- v 1.0)))))
     4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 540000000000.0) {
		tmp = ((3.0 + (2.0 / (r * r))) + (((r * w) * 0.375) * (r * (w / (v - 1.0))))) - 4.5;
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (v - 1.0))))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 540000000000.0d0) then
        tmp = ((3.0d0 + (2.0d0 / (r * r))) + (((r * w) * 0.375d0) * (r * (w / (v - 1.0d0))))) - 4.5d0
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (w * (r / (v - 1.0d0))))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 540000000000.0) {
		tmp = ((3.0 + (2.0 / (r * r))) + (((r * w) * 0.375) * (r * (w / (v - 1.0))))) - 4.5;
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (v - 1.0))))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 540000000000.0:
		tmp = ((3.0 + (2.0 / (r * r))) + (((r * w) * 0.375) * (r * (w / (v - 1.0))))) - 4.5
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (v - 1.0))))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 540000000000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(Float64(Float64(r * w) * 0.375) * Float64(r * Float64(w / Float64(v - 1.0))))) - 4.5);
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(w * Float64(r / Float64(v - 1.0))))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 540000000000.0)
		tmp = ((3.0 + (2.0 / (r * r))) + (((r * w) * 0.375) * (r * (w / (v - 1.0))))) - 4.5;
	else
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (v - 1.0))))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 540000000000.0], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] * N[(r * N[(w / N[(v - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / N[(v - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.4e11

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* 87.0%

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

      2. cancel-sign-sub-inv 87.0%

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

      3. metadata-eval 87.0%

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

      4. +-commutative 87.0%

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

      5. *-commutative 87.0%

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

      6. fma-undefine 87.0%

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

      7. *-commutative 87.0%

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

      8. *-commutative 87.0%

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

      9. associate-/l* 87.0%

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

      10. *-commutative 87.0%

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

      11. associate-*r/ 87.0%

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

      12. associate-*r* 83.6%

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

      13. associate-*l* 92.7%

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

      14. associate-*r* 93.6%

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

   4. Applied egg-rr 93.6%

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

      1. *-commutative 93.6%

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

      2. associate-*r/ 93.6%

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

      3. *-commutative 93.6%

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

      4. associate-/l* 93.4%

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

      5. associate-*l* 95.5%

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

      6. *-commutative 95.5%

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

      7. fma-define 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in v around 0 85.3%

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

      1. *-commutative 85.3%

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

   9. Simplified 85.3%

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

   if 5.4e11 < r

   1. Initial program 91.4%

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

      1. associate--l- 91.4%

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

      2. associate-*l* 82.3%

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

      3. sqr-neg 82.3%

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

      4. associate-*l* 91.4%

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

      5. associate-/l* 96.0%

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

      6. fma-define 96.0%

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

   3. Simplified 96.0%

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

      1. associate-/l* 96.0%

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

      2. *-commutative 96.0%

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

      3. associate-*r/ 95.9%

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

      4. associate-*l* 97.4%

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

      5. associate-*r* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in r around inf 99.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 540000000000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(r \cdot w\right) \cdot 0.375\right) \cdot \left(r \cdot \frac{w}{v - 1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v - 1}\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 100000:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{w \cdot \left(r \cdot 0.375\right)}{\frac{v - 1}{r \cdot w}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 100000.0)
   (- (+ 3.0 (/ 2.0 (* r r))) 4.5)
   (+ 3.0 (- (/ (* w (* r 0.375)) (/ (- v 1.0) (* r w))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 100000.0) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else {
		tmp = 3.0 + (((w * (r * 0.375)) / ((v - 1.0) / (r * w))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 100000.0d0) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - 4.5d0
    else
        tmp = 3.0d0 + (((w * (r * 0.375d0)) / ((v - 1.0d0) / (r * w))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 100000.0) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else {
		tmp = 3.0 + (((w * (r * 0.375)) / ((v - 1.0) / (r * w))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 100000.0:
		tmp = (3.0 + (2.0 / (r * r))) - 4.5
	else:
		tmp = 3.0 + (((w * (r * 0.375)) / ((v - 1.0) / (r * w))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 100000.0)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - 4.5);
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(w * Float64(r * 0.375)) / Float64(Float64(v - 1.0) / Float64(r * w))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 100000.0)
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	else
		tmp = 3.0 + (((w * (r * 0.375)) / ((v - 1.0) / (r * w))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 100000.0], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 + N[(N[(N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(v - 1.0), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1e5

   1. Initial program 84.6%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in r around 0 64.0%

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

   if 1e5 < r

   1. Initial program 91.5%

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

      1. associate--l- 91.5%

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

      2. associate-*l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-*l* 91.5%

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

      5. associate-/l* 96.0%

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

      6. fma-define 96.0%

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

   3. Simplified 96.0%

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

      1. add-sqr-sqrt 95.9%

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

      2. *-un-lft-identity 95.9%

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

      3. times-frac 95.9%

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

      4. associate-*r* 85.5%

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

      5. sqrt-prod 85.5%

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

      6. sqrt-prod 95.9%

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

      7. add-sqr-sqrt 96.0%

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

      8. sqrt-prod 45.9%

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

      9. add-sqr-sqrt 55.8%

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

      10. associate-*r* 48.4%

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

      11. sqrt-prod 48.4%

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

      12. sqrt-prod 55.8%

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

      13. add-sqr-sqrt 55.8%

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

      14. sqrt-prod 46.6%

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

      15. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-*r* 98.3%

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

      3. clear-num 98.2%

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

      4. un-div-inv 98.3%

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

      5. distribute-lft-in 98.3%

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

      6. metadata-eval 98.3%

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

      7. associate-*r* 98.3%

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

      8. metadata-eval 98.3%

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

   9. Applied egg-rr 98.3%

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

   10. Taylor expanded in v around 0 59.8%

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

       1. associate-*r* 59.8%

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

   12. Simplified 59.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 100000:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{w \cdot \left(r \cdot 0.375\right)}{\frac{v - 1}{r \cdot w}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 105:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{1 - v}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 105.0)
   (- (+ 3.0 (/ 2.0 (* r r))) 4.5)
   (- 3.0 (+ 4.5 (/ (* (* r w) 0.375) (/ (- 1.0 v) (* r w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 105.0) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 105.0d0) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - 4.5d0
    else
        tmp = 3.0d0 - (4.5d0 + (((r * w) * 0.375d0) / ((1.0d0 - v) / (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 105.0) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 105.0:
		tmp = (3.0 + (2.0 / (r * r))) - 4.5
	else:
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 105.0)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - 4.5);
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * 0.375) / Float64(Float64(1.0 - v) / Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 105.0)
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	else
		tmp = 3.0 - (4.5 + (((r * w) * 0.375) / ((1.0 - v) / (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 105.0], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 105

   1. Initial program 84.6%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in r around 0 64.0%

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

   if 105 < r

   1. Initial program 91.5%

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

      1. associate--l- 91.5%

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

      2. associate-*l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-*l* 91.5%

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

      5. associate-/l* 96.0%

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

      6. fma-define 96.0%

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

   3. Simplified 96.0%

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

      1. add-sqr-sqrt 95.9%

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

      2. *-un-lft-identity 95.9%

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

      3. times-frac 95.9%

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

      4. associate-*r* 85.5%

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

      5. sqrt-prod 85.5%

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

      6. sqrt-prod 95.9%

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

      7. add-sqr-sqrt 96.0%

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

      8. sqrt-prod 45.9%

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

      9. add-sqr-sqrt 55.8%

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

      10. associate-*r* 48.4%

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

      11. sqrt-prod 48.4%

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

      12. sqrt-prod 55.8%

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

      13. add-sqr-sqrt 55.8%

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

      14. sqrt-prod 46.6%

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

      15. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-*r* 98.3%

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

      3. clear-num 98.2%

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

      4. un-div-inv 98.3%

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

      5. distribute-lft-in 98.3%

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

      6. metadata-eval 98.3%

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

      7. associate-*r* 98.3%

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

      8. metadata-eval 98.3%

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

   9. Applied egg-rr 98.3%

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

   10. Taylor expanded in v around 0 59.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 105:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{1 - v}{r \cdot w}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.5% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Simplified 82.9%

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

5. Taylor expanded in r around 0 53.3%

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

Alternative 13: 13.9% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Simplified 82.9%

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

5. Taylor expanded in r around 0 53.3%

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

6. Taylor expanded in r around inf 11.6%

   \[\leadsto \color{blue}{-1.5} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(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))