Rosa's TurbineBenchmark

Percentage Accurate: 84.4% → 99.8%

Time: 12.6s

Alternatives: 9

Speedup: 1.2×

• 53.3% 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: 84.4% 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: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 + \frac{-0.25 \cdot v + 0.375}{\frac{v + -1}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (+ -1.5 (/ (+ (* -0.25 v) 0.375) (/ (+ v -1.0) (* (* r w) (* r w)))))))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + (((-0.25 * v) + 0.375) / ((v + -1.0) / ((r * w) * (r * w)))));
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (2.0d0 / (r * r)) + ((-1.5d0) + ((((-0.25d0) * v) + 0.375d0) / ((v + (-1.0d0)) / ((r * w) * (r * w)))))
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + (((-0.25 * v) + 0.375) / ((v + -1.0) / ((r * w) * (r * w)))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + (((-0.25 * v) + 0.375) / ((v + -1.0) / ((r * w) * (r * w)))))

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(Float64(-0.25 * v) + 0.375) / Float64(Float64(v + -1.0) / Float64(Float64(r * w) * Float64(r * w))))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + (((-0.25 * v) + 0.375) / ((v + -1.0) / ((r * w) * (r * w)))));
end

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(N[(-0.25 * v), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.9%

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

3. Simplified 89.6%

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

   1. fma-undefine 89.6%

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

   2. *-commutative 89.6%

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

   3. +-commutative 89.6%

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

   4. metadata-eval 89.6%

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

   5. cancel-sign-sub-inv 89.6%

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

   6. associate-*r/ 89.6%

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

   7. *-commutative 89.6%

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

   8. associate-/l* 90.0%

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

   9. clear-num 90.0%

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

   10. un-div-inv 90.0%

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

6. Applied egg-rr 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{-0.25 \cdot v + 0.375}{\frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]
7. Step-by-step derivation

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 + \frac{-0.25 \cdot v + 0.375}{\frac{v + -1}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]
10. Add Preprocessing

Alternative 2: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4.2 \cdot 10^{-93}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 25000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{elif}\;r \leq 4.6 \cdot 10^{+91}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(-0.25 \cdot v + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4.2e-93)
   (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5)
   (if (<= r 25000000.0)
     (+ (/ 2.0 (* r r)) (- -1.5 (* 0.375 (* r (* (* w w) (/ r (- 1.0 v)))))))
     (if (<= r 4.6e+91)
       (-
        (+ 3.0 (* (* w (* r (+ (* -0.25 v) 0.375))) (* w (/ r (+ v -1.0)))))
        4.5)
       (- (- 3.0 (* (* r w) (* r (* w 0.25)))) 4.5)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.2e-93) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 25000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	} else if (r <= 4.6e+91) {
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 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 <= 4.2d-93) then
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - 4.5d0
    else if (r <= 25000000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (0.375d0 * (r * ((w * w) * (r / (1.0d0 - v))))))
    else if (r <= 4.6d+91) then
        tmp = (3.0d0 + ((w * (r * (((-0.25d0) * v) + 0.375d0))) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    else
        tmp = (3.0d0 - ((r * w) * (r * (w * 0.25d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.2e-93) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 25000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	} else if (r <= 4.6e+91) {
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 4.2e-93:
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5
	elif r <= 25000000.0:
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))))
	elif r <= 4.6e+91:
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5
	else:
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 4.2e-93)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - 4.5);
	elseif (r <= 25000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(1.0 - v)))))));
	elseif (r <= 4.6e+91)
		tmp = Float64(Float64(3.0 + Float64(Float64(w * Float64(r * Float64(Float64(-0.25 * v) + 0.375))) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r * w) * Float64(r * Float64(w * 0.25)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 4.2e-93)
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	elseif (r <= 25000000.0)
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	elseif (r <= 4.6e+91)
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	else
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4.2e-93], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 25000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 4.6e+91], N[(N[(3.0 + N[(N[(w * N[(r * N[(N[(-0.25 * v), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if r < 4.2000000000000002e-93

   1. Initial program 85.7%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in r around 0 64.7%

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

      1. associate-/r* 64.8%

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

      2. div-inv 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. associate-*r/ 64.8%

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

      2. *-rgt-identity 64.8%

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

   9. Simplified 64.8%

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

   if 4.2000000000000002e-93 < r < 2.5e7

   1. Initial program 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 2.5e7 < r < 4.59999999999999982e91

   1. Initial program 99.7%

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

   3. Taylor expanded in r around inf 99.7%

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

      1. associate-/l* 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-undefine 99.6%

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

      7. *-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 96.2%

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

      10. *-commutative 96.2%

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

      11. associate-*r/ 96.2%

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

      12. associate-*r* 96.0%

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

      13. associate-*l* 95.9%

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

      14. associate-*r* 96.0%

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

   5. Applied egg-rr 96.0%

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

   if 4.59999999999999982e91 < r

   1. Initial program 91.6%

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

   3. Taylor expanded in r around inf 91.6%

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

   4. Taylor expanded in v around inf 59.7%

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

      1. *-commutative 59.7%

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

   6. Simplified 59.7%

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

      1. associate-/l* 65.2%

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

      2. *-commutative 65.2%

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

      3. *-commutative 65.2%

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

      4. *-commutative 65.2%

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

      5. associate-*r* 56.7%

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

      6. swap-sqr 67.9%

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

      7. associate-*r/ 67.9%

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

      8. *-commutative 67.9%

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

      9. associate-*r* 67.9%

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

      10. *-commutative 67.9%

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

      11. associate-/l* 66.4%

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

   8. Applied egg-rr 66.4%

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

   9. Taylor expanded in v around inf 92.8%

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

       1. *-commutative 92.8%

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

       2. associate-*l* 92.8%

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

   11. Simplified 92.8%

       \[\leadsto \left(3 - \color{blue}{\left(r \cdot \left(w \cdot 0.25\right)\right)} \cdot \left(r \cdot w\right)\right) - 4.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4.2 \cdot 10^{-93}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 25000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{elif}\;r \leq 4.6 \cdot 10^{+91}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(-0.25 \cdot v + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 24000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.1e-92)
   (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5)
   (if (<= r 24000000.0)
     (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0)))))))
     (-
      3.0
      (+
       4.5
       (* (* 0.125 (+ 3.0 (* v -2.0))) (* (* r w) (* w (/ r (- 1.0 v))))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.1e-92) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 24000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w * (r / (1.0 - v))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2.1d-92) then
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - 4.5d0
    else if (r <= 24000000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * ((r * w) * (w * (r / (1.0d0 - v))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.1e-92) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 24000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w * (r / (1.0 - v))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.1e-92:
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5
	elif r <= 24000000.0:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w * (r / (1.0 - v))))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.1e-92)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - 4.5);
	elseif (r <= 24000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(Float64(r * w) * Float64(w * Float64(r / Float64(1.0 - v)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.1e-92)
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	elseif (r <= 24000000.0)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w * (r / (1.0 - v))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.1e-92], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 24000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 2.1e-92

   1. Initial program 85.7%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in r around 0 64.7%

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

      1. associate-/r* 64.8%

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

      2. div-inv 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. associate-*r/ 64.8%

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

      2. *-rgt-identity 64.8%

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

   9. Simplified 64.8%

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

   if 2.1e-92 < r < 2.4e7

   1. Initial program 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 2.4e7 < r

   1. Initial program 95.1%

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

      1. associate--l- 95.1%

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

      2. associate-*l* 88.6%

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

      3. sqr-neg 88.6%

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

      4. associate-*l* 95.1%

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

      5. associate-/l* 98.2%

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

      6. fma-define 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in r around inf 98.2%

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

      1. associate-/l* 96.7%

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

      2. *-commutative 96.7%

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

      3. associate-*r/ 96.7%

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

      4. associate-*l* 98.3%

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

      5. associate-*r* 99.7%

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 24000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 25500000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(w \cdot \left(r \cdot \frac{r \cdot w}{v + -1}\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5e-93)
   (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5)
   (if (<= r 25500000.0)
     (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0)))))))
     (+
      3.0
      (-
       (* (* 0.125 (+ 3.0 (* v -2.0))) (* w (* r (/ (* r w) (+ v -1.0)))))
       4.5)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e-93) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 25500000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * ((r * w) / (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 <= 5d-93) then
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - 4.5d0
    else if (r <= 25500000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * (w * (r * ((r * w) / (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 <= 5e-93) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 25500000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * ((r * w) / (v + -1.0))))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5e-93:
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5
	elif r <= 25500000.0:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * ((r * w) / (v + -1.0))))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5e-93)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - 4.5);
	elseif (r <= 25500000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(w * Float64(r * Float64(Float64(r * w) / Float64(v + -1.0))))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5e-93)
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	elseif (r <= 25500000.0)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * ((r * w) / (v + -1.0))))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5e-93], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 25500000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r * N[(N[(r * w), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 4.99999999999999994e-93

   1. Initial program 85.7%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in r around 0 64.7%

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

      1. associate-/r* 64.8%

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

      2. div-inv 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. associate-*r/ 64.8%

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

      2. *-rgt-identity 64.8%

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

   9. Simplified 64.8%

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

   if 4.99999999999999994e-93 < r < 2.55e7

   1. Initial program 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 2.55e7 < r

   1. Initial program 95.1%

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

      1. associate--l- 95.1%

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

      2. associate-*l* 88.6%

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

      3. sqr-neg 88.6%

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

      4. associate-*l* 95.1%

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

      5. associate-/l* 98.2%

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

      6. fma-define 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in r around inf 98.2%

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

      1. associate-/l* 96.7%

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

      2. *-commutative 96.7%

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

      3. associate-*r/ 96.7%

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

      4. associate-*l* 98.3%

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

      5. associate-*r* 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. pow1 99.7%

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

      2. associate-*l* 98.3%

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

      3. associate-*r/ 98.3%

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

      4. *-commutative 98.3%

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

      5. associate-*r/ 97.5%

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

   9. Applied egg-rr 97.5%

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

       1. unpow1 97.5%

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

       2. *-commutative 97.5%

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

       3. associate-*l* 96.1%

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

       4. associate-*r/ 98.3%

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

   11. Simplified 98.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 25500000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(w \cdot \left(r \cdot \frac{r \cdot w}{v + -1}\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.12 \cdot 10^{-91}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 1.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.12e-91)
   (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5)
   (if (<= r 1.2e+90)
     (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0)))))))
     (- (- 3.0 (* (* r w) (* r (* w 0.25)))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.12e-91) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 1.2e+90) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 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 <= 1.12d-91) then
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - 4.5d0
    else if (r <= 1.2d+90) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = (3.0d0 - ((r * w) * (r * (w * 0.25d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.12e-91) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else if (r <= 1.2e+90) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.12e-91:
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5
	elif r <= 1.2e+90:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.12e-91)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - 4.5);
	elseif (r <= 1.2e+90)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r * w) * Float64(r * Float64(w * 0.25)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.12e-91)
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	elseif (r <= 1.2e+90)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.12e-91], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 1.2e+90], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.12e-91

   1. Initial program 85.7%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in r around 0 64.7%

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

      1. associate-/r* 64.8%

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

      2. div-inv 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. associate-*r/ 64.8%

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

      2. *-rgt-identity 64.8%

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

   9. Simplified 64.8%

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

   if 1.12e-91 < r < 1.20000000000000005e90

   1. Initial program 97.8%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in v around 0 88.0%

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

   if 1.20000000000000005e90 < r

   1. Initial program 91.6%

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

   3. Taylor expanded in r around inf 91.6%

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

   4. Taylor expanded in v around inf 59.7%

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

      1. *-commutative 59.7%

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

   6. Simplified 59.7%

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

      1. associate-/l* 65.2%

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

      2. *-commutative 65.2%

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

      3. *-commutative 65.2%

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

      4. *-commutative 65.2%

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

      5. associate-*r* 56.7%

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

      6. swap-sqr 67.9%

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

      7. associate-*r/ 67.9%

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

      8. *-commutative 67.9%

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

      9. associate-*r* 67.9%

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

      10. *-commutative 67.9%

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

      11. associate-/l* 66.4%

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

   8. Applied egg-rr 66.4%

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

   9. Taylor expanded in v around inf 92.8%

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

       1. *-commutative 92.8%

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

       2. associate-*l* 92.8%

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

   11. Simplified 92.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.12 \cdot 10^{-91}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 1.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 88:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 88.0)
   (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5)
   (- (- 3.0 (* (* r w) (* r (* w 0.25)))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 88.0) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 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 <= 88.0d0) then
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - 4.5d0
    else
        tmp = (3.0d0 - ((r * w) * (r * (w * 0.25d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 88.0) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 88.0:
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5
	else:
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 88.0)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r * w) * Float64(r * Float64(w * 0.25)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 88.0)
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	else
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 88.0], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 88

   1. Initial program 86.9%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in r around 0 66.5%

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

      1. associate-/r* 66.6%

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

      2. div-inv 66.5%

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

   7. Applied egg-rr 66.5%

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

      1. associate-*r/ 66.6%

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

      2. *-rgt-identity 66.6%

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

   9. Simplified 66.6%

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

   if 88 < r

   1. Initial program 95.3%

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

   3. Taylor expanded in r around inf 95.1%

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

   4. Taylor expanded in v around inf 62.3%

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

      1. *-commutative 62.3%

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

   6. Simplified 62.3%

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

      1. associate-/l* 65.3%

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

      2. *-commutative 65.3%

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

      3. *-commutative 65.3%

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

      4. *-commutative 65.3%

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

      5. associate-*r* 60.6%

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

      6. swap-sqr 66.7%

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

      7. associate-*r/ 66.7%

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

      8. *-commutative 66.7%

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

      9. associate-*r* 66.7%

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

      10. *-commutative 66.7%

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

      11. associate-/l* 64.5%

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

   8. Applied egg-rr 64.5%

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

   9. Taylor expanded in v around inf 88.1%

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

       1. *-commutative 88.1%

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

       2. associate-*l* 88.1%

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

   11. Simplified 88.1%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 88:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.9% accurate, 3.2× speedup

Math

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

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(Float64(2.0 / 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[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 88.9%

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

3. Simplified 84.9%

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

5. Taylor expanded in r around 0 55.5%

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

   1. associate-/r* 55.5%

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

   2. div-inv 55.4%

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

7. Applied egg-rr 55.4%

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

   1. associate-*r/ 55.5%

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

   2. *-rgt-identity 55.5%

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

9. Simplified 55.5%

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

Alternative 8: 56.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.9%

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

3. Simplified 84.9%

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

5. Taylor expanded in r around 0 55.5%

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

6. Final simplification 55.5%

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

Alternative 9: 14.2% 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 88.9%

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

3. Simplified 84.9%

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

5. Taylor expanded in r around 0 55.5%

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

6. Taylor expanded in r around inf 12.5%

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

Reproduce

herbie shell --seed 2024140 
(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))