Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 99.8%

Time: 12.1s

Alternatives: 10

Speedup: 1.2×

• 53.4% 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: 84.9% 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

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

FPCore

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

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (-0.25 * v)) / ((1.0 - 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) + ((0.375d0 + ((-0.25d0) * v)) / ((1.0d0 - 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 + ((0.375 + (-0.25 * v)) / ((1.0 - v) * (((-1.0 / r) / w) / (r * w)))));
}

Python

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

MATLAB

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

Derivation

1. Initial program 86.5%

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

3. Simplified 88.2%

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

   1. fma-undefine 88.2%

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

   2. *-commutative 88.2%

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

   3. +-commutative 88.2%

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

   4. metadata-eval 88.2%

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

   5. cancel-sign-sub-inv 88.2%

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

   6. associate-*r/ 88.6%

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

   7. *-commutative 88.6%

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

   8. associate-/l* 89.0%

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

   9. clear-num 89.0%

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

   10. un-div-inv 89.0%

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

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. *-un-lft-identity 99.8%

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

   2. div-inv 99.8%

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

   3. pow-flip 99.8%

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

   4. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

10. Simplified 99.8%

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

    1. sqr-pow 99.8%

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

    2. metadata-eval 99.8%

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

    3. unpow-1 99.8%

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

    4. metadata-eval 99.8%

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

    5. unpow-1 99.8%

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

12. Applied egg-rr 99.8%

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

    1. associate-*r/ 99.8%

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

    2. *-rgt-identity 99.8%

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

    3. associate-/r* 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

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

Alternative 2: 87.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2 \cdot 10^{+22} \lor \neg \left(v \leq 1.5\right):\\ \;\;\;\;t\_1 + \left(-1.5 + \left(-0.25 \cdot v\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 + 0.375 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* (* w w) (/ r (+ v -1.0))))) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -2e+22) (not (<= v 1.5)))
     (+ t_1 (+ -1.5 (* (* -0.25 v) t_0)))
     (+ t_1 (+ -1.5 (* 0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = r * ((w * w) * (r / (v + -1.0)));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2e+22) || !(v <= 1.5)) {
		tmp = t_1 + (-1.5 + ((-0.25 * v) * t_0));
	} else {
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = r * ((w * w) * (r / (v + (-1.0d0))))
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-2d+22)) .or. (.not. (v <= 1.5d0))) then
        tmp = t_1 + ((-1.5d0) + (((-0.25d0) * v) * t_0))
    else
        tmp = t_1 + ((-1.5d0) + (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * ((w * w) * (r / (v + -1.0)));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2e+22) || !(v <= 1.5)) {
		tmp = t_1 + (-1.5 + ((-0.25 * v) * t_0));
	} else {
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * ((w * w) * (r / (v + -1.0)))
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2e+22) or not (v <= 1.5):
		tmp = t_1 + (-1.5 + ((-0.25 * v) * t_0))
	else:
		tmp = t_1 + (-1.5 + (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0))))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2e+22) || !(v <= 1.5))
		tmp = Float64(t_1 + Float64(-1.5 + Float64(Float64(-0.25 * v) * t_0)));
	else
		tmp = Float64(t_1 + Float64(-1.5 + Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * ((w * w) * (r / (v + -1.0)));
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2e+22) || ~((v <= 1.5)))
		tmp = t_1 + (-1.5 + ((-0.25 * v) * t_0));
	else
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -2e+22], N[Not[LessEqual[v, 1.5]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 + N[(N[(-0.25 * v), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 + N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2e22 or 1.5 < v

   1. Initial program 82.7%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in v around inf 86.3%

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

      1. *-commutative 86.3%

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

   7. Simplified 86.3%

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

   if -2e22 < v < 1.5

   1. Initial program 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in v around 0 90.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{0.375} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+22} \lor \neg \left(v \leq 1.5\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(-0.25 \cdot v\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;\left(t\_0 + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 55000000:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 6.2e-59)
     (- (+ t_0 3.0) 4.5)
     (if (<= r 55000000.0)
       (+ t_0 (+ -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 t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.2e-59) {
		tmp = (t_0 + 3.0) - 4.5;
	} else if (r <= 55000000.0) {
		tmp = t_0 + (-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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 6.2d-59) then
        tmp = (t_0 + 3.0d0) - 4.5d0
    else if (r <= 55000000.0d0) then
        tmp = t_0 + ((-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 t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.2e-59) {
		tmp = (t_0 + 3.0) - 4.5;
	} else if (r <= 55000000.0) {
		tmp = t_0 + (-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):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 6.2e-59:
		tmp = (t_0 + 3.0) - 4.5
	elif r <= 55000000.0:
		tmp = t_0 + (-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)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 6.2e-59)
		tmp = Float64(Float64(t_0 + 3.0) - 4.5);
	elseif (r <= 55000000.0)
		tmp = Float64(t_0 + 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)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 6.2e-59)
		tmp = (t_0 + 3.0) - 4.5;
	elseif (r <= 55000000.0)
		tmp = t_0 + (-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_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 6.2e-59], N[(N[(t$95$0 + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 55000000.0], N[(t$95$0 + 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 < 6.19999999999999998e-59

   1. Initial program 84.7%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in r around 0 69.3%

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

   if 6.19999999999999998e-59 < r < 5.5e7

   1. Initial program 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in v around 0 84.6%

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

   if 5.5e7 < r

   1. Initial program 90.9%

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

      1. associate--l- 90.9%

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

      2. associate-*l* 85.1%

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

      3. sqr-neg 85.1%

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

      4. associate-*l* 90.9%

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

      5. associate-/l* 96.0%

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

      6. fma-define 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in r around inf 96.0%

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

      1. associate-/l* 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*r/ 94.7%

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

      4. associate-*l* 97.4%

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

      5. associate-*r* 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 55000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;\left(t\_0 + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 3.4 \cdot 10^{+220}:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 6.2e-59)
     (- (+ t_0 3.0) 4.5)
     (if (<= r 3.4e+220)
       (+ t_0 (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0)))))))
       (- 3.0 (+ 4.5 (* (* 0.125 (+ 3.0 (* 2.0 v))) (* r (* r (* w w))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.2e-59) {
		tmp = (t_0 + 3.0) - 4.5;
	} else if (r <= 3.4e+220) {
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (2.0 * v))) * (r * (r * (w * w)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 6.2d-59) then
        tmp = (t_0 + 3.0d0) - 4.5d0
    else if (r <= 3.4d+220) then
        tmp = t_0 + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + (2.0d0 * v))) * (r * (r * (w * w)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.2e-59) {
		tmp = (t_0 + 3.0) - 4.5;
	} else if (r <= 3.4e+220) {
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (2.0 * v))) * (r * (r * (w * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 6.2e-59:
		tmp = (t_0 + 3.0) - 4.5
	elif r <= 3.4e+220:
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (2.0 * v))) * (r * (r * (w * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 6.2e-59)
		tmp = Float64(Float64(t_0 + 3.0) - 4.5);
	elseif (r <= 3.4e+220)
		tmp = Float64(t_0 + 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(2.0 * v))) * Float64(r * Float64(r * Float64(w * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 6.2e-59)
		tmp = (t_0 + 3.0) - 4.5;
	elseif (r <= 3.4e+220)
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (2.0 * v))) * (r * (r * (w * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 6.2e-59], N[(N[(t$95$0 + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 3.4e+220], N[(t$95$0 + 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[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 6.19999999999999998e-59

   1. Initial program 84.7%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in r around 0 69.3%

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

   if 6.19999999999999998e-59 < r < 3.4e220

   1. Initial program 91.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in v around 0 78.6%

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

   if 3.4e220 < r

   1. Initial program 82.5%

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

      1. associate--l- 82.5%

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

      2. associate-*l* 81.3%

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

      3. sqr-neg 81.3%

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

      4. associate-*l* 82.5%

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

      5. associate-/l* 82.5%

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

      6. fma-define 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in r around inf 82.5%

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

   6. Taylor expanded in v around 0 57.5%

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

      1. metadata-eval 57.5%

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

      2. distribute-lft-neg-in 57.5%

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

      3. neg-sub0 57.5%

         \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \color{blue}{\left(0 - 2 \cdot v\right)}\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]

      4. add-sqr-sqrt 32.5%

         \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \left(0 - \color{blue}{\sqrt{2 \cdot v} \cdot \sqrt{2 \cdot v}}\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]

      5. sqrt-unprod 50.9%

         \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \left(0 - \color{blue}{\sqrt{\left(2 \cdot v\right) \cdot \left(2 \cdot v\right)}}\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]

      6. swap-sqr 50.9%

         \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \left(0 - \sqrt{\color{blue}{\left(2 \cdot 2\right) \cdot \left(v \cdot v\right)}}\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]

      7. metadata-eval 50.9%

         \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \left(0 - \sqrt{\color{blue}{4} \cdot \left(v \cdot v\right)}\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]

      8. metadata-eval 50.9%

         \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \left(0 - \sqrt{\color{blue}{\left(-2 \cdot -2\right)} \cdot \left(v \cdot v\right)}\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]

      9. swap-sqr 50.9%

         \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \left(0 - \sqrt{\color{blue}{\left(-2 \cdot v\right) \cdot \left(-2 \cdot v\right)}}\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]

      10. sqrt-unprod 18.8%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \left(0 - \color{blue}{\sqrt{-2 \cdot v} \cdot \sqrt{-2 \cdot v}}\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]

      11. add-sqr-sqrt 76.2%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \left(0 - \color{blue}{-2 \cdot v}\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]

      12. *-commutative 76.2%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \left(0 - \color{blue}{v \cdot -2}\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]

   8. Applied egg-rr 76.2%

      \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + \color{blue}{\left(0 - v \cdot -2\right)}\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1} + 4.5\right) \]
   9. Step-by-step derivation

      1. neg-sub0 76.2%

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

      2. distribute-rgt-neg-in 76.2%

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

      3. metadata-eval 76.2%

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

      4. *-commutative 76.2%

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

   10. Simplified 76.2%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 3.4 \cdot 10^{+220}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;\left(t\_0 + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 6.2e-59)
     (- (+ t_0 3.0) 4.5)
     (+ t_0 (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.2e-59) {
		tmp = (t_0 + 3.0) - 4.5;
	} else {
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 6.2d-59) then
        tmp = (t_0 + 3.0d0) - 4.5d0
    else
        tmp = t_0 + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.2e-59) {
		tmp = (t_0 + 3.0) - 4.5;
	} else {
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 6.2e-59:
		tmp = (t_0 + 3.0) - 4.5
	else:
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 6.2e-59)
		tmp = Float64(Float64(t_0 + 3.0) - 4.5);
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 6.2e-59)
		tmp = (t_0 + 3.0) - 4.5;
	else
		tmp = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 6.2e-59], N[(N[(t$95$0 + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 + 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]]]

Derivation

1. Split input into 2 regimes

2. if r < 6.19999999999999998e-59

   1. Initial program 84.7%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in r around 0 69.3%

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

   if 6.19999999999999998e-59 < r

   1. Initial program 90.0%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in v around 0 74.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{0.375} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (-0.25 * v)) / ((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) + ((0.375d0 + ((-0.25d0) * v)) / ((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 + ((0.375 + (-0.25 * v)) / ((v + -1.0) / ((r * w) * (r * w)))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (-0.25 * v)) / ((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(0.375 + Float64(-0.25 * v)) / 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 + ((0.375 + (-0.25 * v)) / ((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[(0.375 + N[(-0.25 * v), $MachinePrecision]), $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 86.5%

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

3. Simplified 88.2%

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

   1. fma-undefine 88.2%

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

   2. *-commutative 88.2%

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

   3. +-commutative 88.2%

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

   4. metadata-eval 88.2%

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

   5. cancel-sign-sub-inv 88.2%

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

   6. associate-*r/ 88.6%

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

   7. *-commutative 88.6%

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

   8. associate-/l* 89.0%

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

   9. clear-num 89.0%

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

   10. un-div-inv 89.0%

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

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%

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

Alternative 7: 67.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 13600:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(3 + v \cdot -2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 13600.0)
   (- (+ (/ 2.0 (* r r)) 3.0) 4.5)
   (- 3.0 (+ 4.5 (* (* (* r w) (* r w)) (* 0.125 (+ 3.0 (* v -2.0))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 13600.0) {
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))));
	}
	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 <= 13600.0d0) then
        tmp = ((2.0d0 / (r * r)) + 3.0d0) - 4.5d0
    else
        tmp = 3.0d0 - (4.5d0 + (((r * w) * (r * w)) * (0.125d0 * (3.0d0 + (v * (-2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 13600.0) {
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 13600.0:
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5
	else:
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 13600.0)
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - 4.5);
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * Float64(r * w)) * Float64(0.125 * Float64(3.0 + Float64(v * -2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 13600.0)
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	else
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 13600.0], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 13600

   1. Initial program 84.6%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in r around 0 68.3%

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

   if 13600 < r

   1. Initial program 91.1%

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

      1. associate--l- 91.1%

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

      2. associate-*l* 85.5%

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

      3. sqr-neg 85.5%

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

      4. associate-*l* 91.1%

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

      5. associate-/l* 96.1%

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

      6. fma-define 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in r around inf 96.1%

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

      1. associate-/l* 94.9%

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

      2. *-commutative 94.9%

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

      3. associate-*r/ 94.8%

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

      4. associate-*l* 97.4%

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

      5. associate-*r* 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in v around 0 74.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 13600:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(3 + v \cdot -2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.6% 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 86.5%

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

3. Simplified 82.3%

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

5. Taylor expanded in r around 0 56.7%

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

6. Final simplification 56.7%

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

Alternative 9: 57.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ -1.5 + 2 \cdot \frac{\frac{1}{r}}{r} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

3. Simplified 82.3%

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

5. Taylor expanded in r around 0 56.7%

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

   1. associate--l+ 56.7%

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

   2. div-inv 56.7%

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

   3. fma-neg 56.7%

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

   4. pow2 56.7%

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

   5. pow-flip 56.8%

      \[\leadsto 3 + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -4.5\right) \]

   6. metadata-eval 56.8%

      \[\leadsto 3 + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -4.5\right) \]

   7. metadata-eval 56.8%

      \[\leadsto 3 + \mathsf{fma}\left(2, {r}^{-2}, \color{blue}{-4.5}\right) \]

7. Applied egg-rr 56.8%

   \[\leadsto \color{blue}{3 + \mathsf{fma}\left(2, {r}^{-2}, -4.5\right)} \]
8. Step-by-step derivation

   1. +-commutative 56.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, {r}^{-2}, -4.5\right) + 3} \]

   2. fma-undefine 56.8%

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

   3. associate-+l+ 56.8%

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

   4. metadata-eval 56.8%

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

9. Simplified 56.8%

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

    1. sqr-pow 56.6%

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

    2. metadata-eval 56.6%

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

    3. unpow-1 56.6%

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

    4. metadata-eval 56.6%

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

    5. unpow-1 56.6%

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

11. Applied egg-rr 56.6%

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

    1. associate-*l/ 56.7%

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

    2. *-lft-identity 56.7%

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

13. Simplified 56.7%

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

14. Final simplification 56.7%

    \[\leadsto -1.5 + 2 \cdot \frac{\frac{1}{r}}{r} \]
15. Add Preprocessing

Alternative 10: 13.9% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

3. Simplified 82.3%

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

5. Taylor expanded in r around 0 56.7%

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

6. Taylor expanded in r around inf 16.5%

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

Reproduce

herbie shell --seed 2024146 
(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))