Rosa's TurbineBenchmark

Percentage Accurate: 84.2% → 99.8%

Time: 11.4s

Alternatives: 11

Speedup: 1.2×

• 53.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.2% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

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(1.0 - v) / Float64(Float64(r * w) / Float64(Float64(1.0 / w) / r))))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 - (((-0.25 * v) + 0.375) / ((1.0 - v) / ((r * w) / ((1.0 / w) / r)))));
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[(1.0 - v), $MachinePrecision] / N[(N[(r * w), $MachinePrecision] / N[(N[(1.0 / w), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.6%

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

3. Simplified 86.0%

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

   1. fma-undefine 86.0%

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

   2. *-commutative 86.0%

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

   3. +-commutative 86.0%

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

   4. metadata-eval 86.0%

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

   5. cancel-sign-sub-inv 86.0%

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

   6. associate-*r/ 86.2%

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

   7. *-commutative 86.2%

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

   8. associate-/l* 86.5%

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

   9. clear-num 86.5%

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

   10. un-div-inv 86.5%

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

6. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

8. Applied egg-rr 99.7%

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

   1. pow2 99.7%

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

   2. metadata-eval 99.7%

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

   3. pow-div 99.8%

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

   4. pow1 99.8%

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

   5. inv-pow 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-/r* 99.8%

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

10. Applied egg-rr 99.8%

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

Alternative 2: 74.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.3 \cdot 10^{-19}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 2.45 \cdot 10^{+204}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(-0.25 \cdot v + 0.375\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{1}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.3e-19)
   (- (+ (/ 2.0 (* r r)) 3.0) 4.5)
   (if (<= r 2.45e+204)
     (-
      3.0
      (+
       4.5
       (* (* 0.125 (+ 3.0 (* v -2.0))) (/ (* r (* r (* w w))) (- 1.0 v)))))
     (-
      (+
       3.0
       (* (* (+ (* -0.25 v) 0.375) (* (* r w) (* r w))) (/ 1.0 (+ v -1.0))))
      4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.3e-19) {
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	} else if (r <= 2.45e+204) {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * (r * (w * w))) / (1.0 - v))));
	} else {
		tmp = (3.0 + ((((-0.25 * v) + 0.375) * ((r * w) * (r * w))) * (1.0 / (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 <= 1.3d-19) then
        tmp = ((2.0d0 / (r * r)) + 3.0d0) - 4.5d0
    else if (r <= 2.45d+204) then
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * ((r * (r * (w * w))) / (1.0d0 - v))))
    else
        tmp = (3.0d0 + (((((-0.25d0) * v) + 0.375d0) * ((r * w) * (r * w))) * (1.0d0 / (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 <= 1.3e-19) {
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	} else if (r <= 2.45e+204) {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * (r * (w * w))) / (1.0 - v))));
	} else {
		tmp = (3.0 + ((((-0.25 * v) + 0.375) * ((r * w) * (r * w))) * (1.0 / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.3e-19:
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5
	elif r <= 2.45e+204:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * (r * (w * w))) / (1.0 - v))))
	else:
		tmp = (3.0 + ((((-0.25 * v) + 0.375) * ((r * w) * (r * w))) * (1.0 / (v + -1.0)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.3e-19)
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - 4.5);
	elseif (r <= 2.45e+204)
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(1.0 - v)))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(Float64(-0.25 * v) + 0.375) * Float64(Float64(r * w) * Float64(r * w))) * Float64(1.0 / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.3e-19)
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	elseif (r <= 2.45e+204)
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * (r * (w * w))) / (1.0 - v))));
	else
		tmp = (3.0 + ((((-0.25 * v) + 0.375) * ((r * w) * (r * w))) * (1.0 / (v + -1.0)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.3e-19], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 2.45e+204], N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(N[(-0.25 * v), $MachinePrecision] + 0.375), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.30000000000000006e-19

   1. Initial program 82.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in r around 0 70.7%

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

   if 1.30000000000000006e-19 < r < 2.4499999999999999e204

   1. Initial program 89.4%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in r around inf 96.3%

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

   if 2.4499999999999999e204 < r

   1. Initial program 81.7%

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

      1. div-inv 81.6%

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

      2. *-commutative 81.6%

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

      3. *-commutative 81.6%

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

      4. *-commutative 81.6%

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

      5. associate-*r* 65.6%

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

      6. pow2 65.6%

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

      7. pow2 65.6%

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

      8. pow-prod-down 99.8%

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

      9. cancel-sign-sub-inv 99.8%

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

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

      12. distribute-lft-in 99.8%

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

      13. metadata-eval 99.8%

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

      14. associate-*r* 99.8%

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

      15. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow2 99.6%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in r around inf 99.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.3 \cdot 10^{-19}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 2.45 \cdot 10^{+204}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(-0.25 \cdot v + 0.375\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{1}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3.4 \cdot 10^{-19}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 8.5 \cdot 10^{+220}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(-0.25 \cdot v + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 3.4e-19)
   (- (+ (/ 2.0 (* r r)) 3.0) 4.5)
   (if (<= r 8.5e+220)
     (-
      3.0
      (+
       4.5
       (* (* 0.125 (+ 3.0 (* v -2.0))) (/ (* r (* r (* w w))) (- 1.0 v)))))
     (-
      (+ 3.0 (* (* w (* r (+ (* -0.25 v) 0.375))) (* w (/ r (+ v -1.0)))))
      4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.4e-19) {
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	} else if (r <= 8.5e+220) {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * (r * (w * w))) / (1.0 - v))));
	} else {
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (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 <= 3.4d-19) then
        tmp = ((2.0d0 / (r * r)) + 3.0d0) - 4.5d0
    else if (r <= 8.5d+220) then
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * ((r * (r * (w * w))) / (1.0d0 - v))))
    else
        tmp = (3.0d0 + ((w * (r * (((-0.25d0) * v) + 0.375d0))) * (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 <= 3.4e-19) {
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	} else if (r <= 8.5e+220) {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * (r * (w * w))) / (1.0 - v))));
	} else {
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 3.4e-19:
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5
	elif r <= 8.5e+220:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * (r * (w * w))) / (1.0 - v))))
	else:
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3.4e-19)
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - 4.5);
	elseif (r <= 8.5e+220)
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(1.0 - v)))));
	else
		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);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 3.4e-19)
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	elseif (r <= 8.5e+220)
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (v * -2.0))) * ((r * (r * (w * w))) / (1.0 - v))));
	else
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3.4e-19], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 8.5e+220], N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if r < 3.4000000000000002e-19

   1. Initial program 82.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in r around 0 70.7%

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

   if 3.4000000000000002e-19 < r < 8.4999999999999996e220

   1. Initial program 90.3%

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

   3. Simplified 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 96.5%

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

   if 8.4999999999999996e220 < r

   1. Initial program 75.7%

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

   3. Taylor expanded in r around inf 75.7%

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

      1. associate-/l* 75.7%

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

      2. cancel-sign-sub-inv 75.7%

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

      3. metadata-eval 75.7%

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

      4. +-commutative 75.7%

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

      5. *-commutative 75.7%

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

      6. fma-undefine 75.7%

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

      7. *-commutative 75.7%

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

      8. *-commutative 75.7%

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

      9. associate-/l* 75.7%

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

      10. *-commutative 75.7%

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

      11. associate-*r/ 75.7%

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

      12. associate-*r* 75.2%

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

      13. associate-*l* 93.1%

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

      14. associate-*r* 93.4%

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

   5. Applied egg-rr 93.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.4 \cdot 10^{-19}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 8.5 \cdot 10^{+220}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(-0.25 \cdot v + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 3.5 \cdot 10^{-88}:\\ \;\;\;\;\left(t\_0 + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 64000000:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(-0.25 \cdot v + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 3.5e-88)
     (- (+ t_0 3.0) 4.5)
     (if (<= r 64000000.0)
       (+ t_0 (- -1.5 (* 0.375 (* r (* (* w w) (/ r (- 1.0 v)))))))
       (-
        (+ 3.0 (* (* w (* r (+ (* -0.25 v) 0.375))) (* w (/ r (+ v -1.0)))))
        4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3.5e-88) {
		tmp = (t_0 + 3.0) - 4.5;
	} else if (r <= 64000000.0) {
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 3.5d-88) then
        tmp = (t_0 + 3.0d0) - 4.5d0
    else if (r <= 64000000.0d0) then
        tmp = t_0 + ((-1.5d0) - (0.375d0 * (r * ((w * w) * (r / (1.0d0 - v))))))
    else
        tmp = (3.0d0 + ((w * (r * (((-0.25d0) * v) + 0.375d0))) * (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 t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3.5e-88) {
		tmp = (t_0 + 3.0) - 4.5;
	} else if (r <= 64000000.0) {
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 3.5e-88:
		tmp = (t_0 + 3.0) - 4.5
	elif r <= 64000000.0:
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))))
	else:
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 3.5e-88)
		tmp = Float64(Float64(t_0 + 3.0) - 4.5);
	elseif (r <= 64000000.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(1.0 - v)))))));
	else
		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);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 3.5e-88)
		tmp = (t_0 + 3.0) - 4.5;
	elseif (r <= 64000000.0)
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	else
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (w * (r / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 3.5e-88], N[(N[(t$95$0 + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 64000000.0], N[(t$95$0 + 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], 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]]]]

Derivation

1. Split input into 3 regimes

2. if r < 3.5000000000000001e-88

   1. Initial program 82.9%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in r around 0 70.8%

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

   if 3.5000000000000001e-88 < r < 6.4e7

   1. Initial program 71.1%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in v around 0 70.1%

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

   if 6.4e7 < r

   1. Initial program 88.0%

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

   3. Taylor expanded in r around inf 88.0%

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

      1. associate-/l* 93.4%

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

      2. cancel-sign-sub-inv 93.4%

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

      3. metadata-eval 93.4%

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

      4. +-commutative 93.4%

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

      5. *-commutative 93.4%

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

      6. fma-undefine 93.4%

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

      7. *-commutative 93.4%

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

      8. *-commutative 93.4%

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

      9. associate-/l* 92.1%

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

      10. *-commutative 92.1%

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

      11. associate-*r/ 92.0%

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

      12. associate-*r* 82.1%

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

      13. associate-*l* 85.9%

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

      14. associate-*r* 86.0%

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

   5. Applied egg-rr 86.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.5 \cdot 10^{-88}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 64000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(-0.25 \cdot v + 0.375\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 1.25 \cdot 10^{-93}:\\ \;\;\;\;\left(t\_0 + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 86000000:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(-0.25 \cdot v + 0.375\right)\right)\right) \cdot \left(r \cdot \frac{w}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 1.25e-93)
     (- (+ t_0 3.0) 4.5)
     (if (<= r 86000000.0)
       (+ t_0 (- -1.5 (* 0.375 (* r (* (* w w) (/ r (- 1.0 v)))))))
       (-
        (+ 3.0 (* (* w (* r (+ (* -0.25 v) 0.375))) (* r (/ w (+ v -1.0)))))
        4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1.25e-93) {
		tmp = (t_0 + 3.0) - 4.5;
	} else if (r <= 86000000.0) {
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 1.25d-93) then
        tmp = (t_0 + 3.0d0) - 4.5d0
    else if (r <= 86000000.0d0) then
        tmp = t_0 + ((-1.5d0) - (0.375d0 * (r * ((w * w) * (r / (1.0d0 - v))))))
    else
        tmp = (3.0d0 + ((w * (r * (((-0.25d0) * v) + 0.375d0))) * (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 t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1.25e-93) {
		tmp = (t_0 + 3.0) - 4.5;
	} else if (r <= 86000000.0) {
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (r * (w / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 1.25e-93:
		tmp = (t_0 + 3.0) - 4.5
	elif r <= 86000000.0:
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))))
	else:
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (r * (w / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 1.25e-93)
		tmp = Float64(Float64(t_0 + 3.0) - 4.5);
	elseif (r <= 86000000.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(1.0 - v)))))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(w * Float64(r * Float64(Float64(-0.25 * v) + 0.375))) * Float64(r * Float64(w / Float64(v + -1.0))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 1.25e-93)
		tmp = (t_0 + 3.0) - 4.5;
	elseif (r <= 86000000.0)
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	else
		tmp = (3.0 + ((w * (r * ((-0.25 * v) + 0.375))) * (r * (w / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.25e-93], N[(N[(t$95$0 + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 86000000.0], N[(t$95$0 + 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], N[(N[(3.0 + N[(N[(w * N[(r * N[(N[(-0.25 * v), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.24999999999999999e-93

   1. Initial program 82.9%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in r around 0 70.8%

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

   if 1.24999999999999999e-93 < r < 8.6e7

   1. Initial program 71.1%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in v around 0 70.1%

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

   if 8.6e7 < r

   1. Initial program 88.0%

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

   3. Taylor expanded in r around inf 88.0%

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

      1. associate-/l* 93.4%

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

      2. cancel-sign-sub-inv 93.4%

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

      3. metadata-eval 93.4%

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

      4. +-commutative 93.4%

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

      5. *-commutative 93.4%

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

      6. fma-undefine 93.4%

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

      7. *-commutative 93.4%

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

      8. *-commutative 93.4%

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

      9. associate-/l* 92.1%

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

      10. *-commutative 92.1%

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

      11. associate-*r/ 92.0%

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

      12. associate-*r* 82.1%

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

      13. associate-*l* 85.9%

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

      14. associate-*r* 86.0%

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

   5. Applied egg-rr 86.0%

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

   6. Taylor expanded in w around 0 86.1%

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

      1. associate-/l* 85.9%

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

   8. Simplified 85.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.25 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 86000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(-0.25 \cdot v + 0.375\right)\right)\right) \cdot \left(r \cdot \frac{w}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.1 \cdot 10^{-91}:\\ \;\;\;\;\left(t\_0 + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 68000000:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.1e-91)
     (- (+ t_0 3.0) 4.5)
     (if (<= r 68000000.0)
       (+ t_0 (- -1.5 (* 0.375 (* r (* (* w w) (/ r (- 1.0 v)))))))
       (- (- 3.0 (* 0.375 (* r (* r (* w w))))) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.1e-91) {
		tmp = (t_0 + 3.0) - 4.5;
	} else if (r <= 68000000.0) {
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = (3.0 - (0.375 * (r * (r * (w * 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 2.1d-91) then
        tmp = (t_0 + 3.0d0) - 4.5d0
    else if (r <= 68000000.0d0) then
        tmp = t_0 + ((-1.5d0) - (0.375d0 * (r * ((w * w) * (r / (1.0d0 - v))))))
    else
        tmp = (3.0d0 - (0.375d0 * (r * (r * (w * w))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.1e-91) {
		tmp = (t_0 + 3.0) - 4.5;
	} else if (r <= 68000000.0) {
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = (3.0 - (0.375 * (r * (r * (w * w))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.1e-91:
		tmp = (t_0 + 3.0) - 4.5
	elif r <= 68000000.0:
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))))
	else:
		tmp = (3.0 - (0.375 * (r * (r * (w * w))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.1e-91)
		tmp = Float64(Float64(t_0 + 3.0) - 4.5);
	elseif (r <= 68000000.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(1.0 - v)))))));
	else
		tmp = Float64(Float64(3.0 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.1e-91)
		tmp = (t_0 + 3.0) - 4.5;
	elseif (r <= 68000000.0)
		tmp = t_0 + (-1.5 - (0.375 * (r * ((w * w) * (r / (1.0 - v))))));
	else
		tmp = (3.0 - (0.375 * (r * (r * (w * w))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.1e-91], N[(N[(t$95$0 + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 68000000.0], N[(t$95$0 + 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], N[(N[(3.0 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 2.0999999999999999e-91

   1. Initial program 82.9%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in r around 0 70.8%

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

   if 2.0999999999999999e-91 < r < 6.8e7

   1. Initial program 71.1%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in v around 0 70.1%

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

   if 6.8e7 < r

   1. Initial program 88.0%

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

   3. Taylor expanded in r around inf 88.0%

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

   4. Taylor expanded in v around 0 66.6%

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

   5. Taylor expanded in v around 0 81.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.1 \cdot 10^{-91}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 68000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 - (((-0.25 * v) + 0.375) / (((1.0 / w) / r) * ((1.0 - v) / (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[(N[(1.0 / w), $MachinePrecision] / r), $MachinePrecision] * N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.6%

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

3. Simplified 86.0%

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

   1. fma-undefine 86.0%

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

   2. *-commutative 86.0%

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

   3. +-commutative 86.0%

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

   4. metadata-eval 86.0%

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

   5. cancel-sign-sub-inv 86.0%

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

   6. associate-*r/ 86.2%

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

   7. *-commutative 86.2%

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

   8. associate-/l* 86.5%

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

   9. clear-num 86.5%

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

   10. un-div-inv 86.5%

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

6. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

   2. /-rgt-identity 99.7%

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

   3. /-rgt-identity 99.7%

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

   4. clear-num 99.7%

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

   5. frac-times 99.8%

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

   6. metadata-eval 99.8%

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

   7. div-inv 99.8%

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

   8. /-rgt-identity 99.8%

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

   9. *-un-lft-identity 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-/r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-/r* 99.8%

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

10. Applied egg-rr 99.8%

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

Alternative 8: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (- -1.5 (/ (+ (* -0.25 v) 0.375) (/ (- 1.0 v) (* (* 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) / ((1.0 - v) / ((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) / ((1.0d0 - v) / ((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) / ((1.0 - v) / ((r * w) * (r * w)))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 - (((-0.25 * v) + 0.375) / ((1.0 - v) / ((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(1.0 - v) / 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) / ((1.0 - v) / ((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[(1.0 - v), $MachinePrecision] / N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.6%

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

3. Simplified 86.0%

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

   1. fma-undefine 86.0%

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

   2. *-commutative 86.0%

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

   3. +-commutative 86.0%

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

   4. metadata-eval 86.0%

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

   5. cancel-sign-sub-inv 86.0%

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

   6. associate-*r/ 86.2%

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

   7. *-commutative 86.2%

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

   8. associate-/l* 86.5%

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

   9. clear-num 86.5%

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

   10. un-div-inv 86.5%

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

6. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 9: 71.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 60000000:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 60000000.0)
   (- (+ (/ 2.0 (* r r)) 3.0) 4.5)
   (- (- 3.0 (* 0.375 (* r (* r (* w w))))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 60000000.0) {
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	} else {
		tmp = (3.0 - (0.375 * (r * (r * (w * 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 <= 60000000.0d0) then
        tmp = ((2.0d0 / (r * r)) + 3.0d0) - 4.5d0
    else
        tmp = (3.0d0 - (0.375d0 * (r * (r * (w * w))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 60000000.0) {
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	} else {
		tmp = (3.0 - (0.375 * (r * (r * (w * w))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 60000000.0:
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5
	else:
		tmp = (3.0 - (0.375 * (r * (r * (w * w))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 60000000.0)
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 60000000.0)
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	else
		tmp = (3.0 - (0.375 * (r * (r * (w * w))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 60000000.0], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 6e7

   1. Initial program 81.9%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in r around 0 70.1%

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

   if 6e7 < r

   1. Initial program 88.0%

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

   3. Taylor expanded in r around inf 88.0%

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

   4. Taylor expanded in v around 0 66.6%

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

   5. Taylor expanded in v around 0 81.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 60000000:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.3% 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 83.6%

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

3. Simplified 79.3%

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

5. Taylor expanded in r around 0 58.0%

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

6. Final simplification 58.0%

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

Alternative 11: 14.5% 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 83.6%

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

3. Simplified 79.3%

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

5. Taylor expanded in r around 0 58.0%

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

6. Taylor expanded in r around inf 16.5%

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

Reproduce

herbie shell --seed 2024180 
(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))