Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.8%

Time: 12.1s

Alternatives: 12

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: 84.8% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 83.9%

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

3. Simplified 85.4%

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

   1. *-commutative 85.4%

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

   2. fma-undefine 85.4%

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

   3. *-commutative 85.4%

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

   4. +-commutative 85.4%

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

   5. metadata-eval 85.4%

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

   6. cancel-sign-sub-inv 85.4%

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

   7. associate-*r/ 85.4%

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

   8. *-commutative 85.4%

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

   9. associate-/l* 85.8%

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

   10. *-commutative 85.8%

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

   11. clear-num 85.8%

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

   12. un-div-inv 85.8%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. unpow2 99.8%

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

   3. times-frac 99.8%

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

8. Applied egg-rr 99.8%

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

   1. inv-pow 99.8%

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

   2. unpow-prod-down 99.8%

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

   3. inv-pow 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.8%

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

    2. *-lft-identity 99.8%

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

    3. unpow-1 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 0.375 + v \cdot -0.25\\ \mathbf{if}\;r \leq 1.2 \cdot 10^{-95}:\\ \;\;\;\;t\_0 + \left(-1.5 + \frac{t\_1}{\frac{\frac{1}{w}}{r} \cdot \frac{\frac{-1}{r}}{w}}\right)\\ \mathbf{elif}\;r \leq 1.82 \cdot 10^{+50}:\\ \;\;\;\;t\_0 + \left(-1.5 - t\_1 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(w \cdot \left(r \cdot \left(w \cdot \frac{r}{v + -1}\right)\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ 0.375 (* v -0.25))))
   (if (<= r 1.2e-95)
     (+ t_0 (+ -1.5 (/ t_1 (* (/ (/ 1.0 w) r) (/ (/ -1.0 r) w)))))
     (if (<= r 1.82e+50)
       (+ t_0 (- -1.5 (* t_1 (* r (* (* w w) (/ r (- 1.0 v)))))))
       (+
        3.0
        (-
         (* (* 0.125 (+ 3.0 (* v -2.0))) (* w (* r (* w (/ r (+ v -1.0))))))
         4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 0.375 + (v * -0.25);
	double tmp;
	if (r <= 1.2e-95) {
		tmp = t_0 + (-1.5 + (t_1 / (((1.0 / w) / r) * ((-1.0 / r) / w))));
	} else if (r <= 1.82e+50) {
		tmp = t_0 + (-1.5 - (t_1 * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * (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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = 0.375d0 + (v * (-0.25d0))
    if (r <= 1.2d-95) then
        tmp = t_0 + ((-1.5d0) + (t_1 / (((1.0d0 / w) / r) * (((-1.0d0) / r) / w))))
    else if (r <= 1.82d+50) then
        tmp = t_0 + ((-1.5d0) - (t_1 * (r * ((w * w) * (r / (1.0d0 - v))))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * (w * (r * (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 t_1 = 0.375 + (v * -0.25);
	double tmp;
	if (r <= 1.2e-95) {
		tmp = t_0 + (-1.5 + (t_1 / (((1.0 / w) / r) * ((-1.0 / r) / w))));
	} else if (r <= 1.82e+50) {
		tmp = t_0 + (-1.5 - (t_1 * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * (w * (r / (v + -1.0)))))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = 0.375 + (v * -0.25)
	tmp = 0
	if r <= 1.2e-95:
		tmp = t_0 + (-1.5 + (t_1 / (((1.0 / w) / r) * ((-1.0 / r) / w))))
	elif r <= 1.82e+50:
		tmp = t_0 + (-1.5 - (t_1 * (r * ((w * w) * (r / (1.0 - v))))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * (w * (r / (v + -1.0)))))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(0.375 + Float64(v * -0.25))
	tmp = 0.0
	if (r <= 1.2e-95)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(t_1 / Float64(Float64(Float64(1.0 / w) / r) * Float64(Float64(-1.0 / r) / w)))));
	elseif (r <= 1.82e+50)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(1.0 - v)))))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(w * Float64(r * 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);
	t_1 = 0.375 + (v * -0.25);
	tmp = 0.0;
	if (r <= 1.2e-95)
		tmp = t_0 + (-1.5 + (t_1 / (((1.0 / w) / r) * ((-1.0 / r) / w))));
	elseif (r <= 1.82e+50)
		tmp = t_0 + (-1.5 - (t_1 * (r * ((w * w) * (r / (1.0 - v))))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * (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]}, Block[{t$95$1 = N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.2e-95], N[(t$95$0 + N[(-1.5 + N[(t$95$1 / N[(N[(N[(1.0 / w), $MachinePrecision] / r), $MachinePrecision] * N[(N[(-1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.82e+50], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.2e-95

   1. Initial program 79.3%

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

   3. Simplified 80.4%

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

      1. *-commutative 80.4%

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

      2. fma-undefine 80.4%

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

      3. *-commutative 80.4%

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

      4. +-commutative 80.4%

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

      5. metadata-eval 80.4%

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

      6. cancel-sign-sub-inv 80.4%

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

      7. associate-*r/ 80.4%

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

      8. *-commutative 80.4%

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

      9. associate-/l* 81.0%

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

      10. *-commutative 81.0%

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

      11. clear-num 81.0%

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

      12. un-div-inv 81.0%

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

   6. Applied egg-rr 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. inv-pow 99.8%

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

      2. unpow-prod-down 99.9%

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

      3. inv-pow 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. associate-*l/ 99.9%

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

       2. *-lft-identity 99.9%

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

       3. unpow-1 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in v around 0 81.0%

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

       1. associate-/r* 81.0%

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

   15. Simplified 81.0%

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

   if 1.2e-95 < r < 1.81999999999999997e50

   1. Initial program 93.8%

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

   3. Simplified 96.9%

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

      1. fma-undefine 96.9%

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

      2. *-commutative 96.9%

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

      3. distribute-rgt-in 96.9%

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

      4. *-commutative 96.9%

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

      5. associate-*l* 96.9%

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

      6. metadata-eval 96.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      7. metadata-eval 96.9%

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

   6. Applied egg-rr 96.9%

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

   if 1.81999999999999997e50 < r

   1. Initial program 93.3%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in r around inf 95.1%

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

      1. associate-/l* 95.1%

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

      2. *-commutative 95.1%

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

      3. associate-*r/ 95.1%

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

      4. *-commutative 95.1%

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

      5. associate-*l* 99.8%

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

      6. associate-*l* 98.0%

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

   7. Applied egg-rr 98.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{0.375 + v \cdot -0.25}{\frac{\frac{1}{w}}{r} \cdot \frac{\frac{-1}{r}}{w}}\right)\\ \mathbf{elif}\;r \leq 1.82 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(w \cdot \left(r \cdot \left(w \cdot \frac{r}{v + -1}\right)\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 8.5 \cdot 10^{-112}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{elif}\;r \leq 2.7 \cdot 10^{+50}:\\ \;\;\;\;t\_0 + \left(-1.5 - \left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(w \cdot \left(r \cdot \left(w \cdot \frac{r}{v + -1}\right)\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 8.5e-112)
     (+ t_0 -1.5)
     (if (<= r 2.7e+50)
       (+
        t_0
        (- -1.5 (* (+ 0.375 (* v -0.25)) (* r (* (* w w) (/ r (- 1.0 v)))))))
       (+
        3.0
        (-
         (* (* 0.125 (+ 3.0 (* v -2.0))) (* w (* r (* 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 <= 8.5e-112) {
		tmp = t_0 + -1.5;
	} else if (r <= 2.7e+50) {
		tmp = t_0 + (-1.5 - ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * (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 <= 8.5d-112) then
        tmp = t_0 + (-1.5d0)
    else if (r <= 2.7d+50) then
        tmp = t_0 + ((-1.5d0) - ((0.375d0 + (v * (-0.25d0))) * (r * ((w * w) * (r / (1.0d0 - v))))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * (w * (r * (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 <= 8.5e-112) {
		tmp = t_0 + -1.5;
	} else if (r <= 2.7e+50) {
		tmp = t_0 + (-1.5 - ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * (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 <= 8.5e-112:
		tmp = t_0 + -1.5
	elif r <= 2.7e+50:
		tmp = t_0 + (-1.5 - ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (1.0 - v))))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * (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 <= 8.5e-112)
		tmp = Float64(t_0 + -1.5);
	elseif (r <= 2.7e+50)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(1.0 - v)))))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(w * Float64(r * 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 <= 8.5e-112)
		tmp = t_0 + -1.5;
	elseif (r <= 2.7e+50)
		tmp = t_0 + (-1.5 - ((0.375 + (v * -0.25)) * (r * ((w * w) * (r / (1.0 - v))))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * (r * (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, 8.5e-112], N[(t$95$0 + -1.5), $MachinePrecision], If[LessEqual[r, 2.7e+50], N[(t$95$0 + N[(-1.5 - N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 8.49999999999999992e-112

   1. Initial program 79.9%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in v around inf 78.4%

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

      1. *-commutative 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in r around 0 65.1%

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

   if 8.49999999999999992e-112 < r < 2.7e50

   1. Initial program 89.3%

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

   3. Simplified 92.1%

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

      1. fma-undefine 92.1%

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

      2. *-commutative 92.1%

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

      3. distribute-rgt-in 92.1%

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

      4. *-commutative 92.1%

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

      5. associate-*l* 92.1%

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

      6. metadata-eval 92.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot \color{blue}{-0.25} + 3 \cdot 0.125\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      7. metadata-eval 92.1%

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

   6. Applied egg-rr 92.1%

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

   if 2.7e50 < r

   1. Initial program 93.3%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in r around inf 95.1%

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

      1. associate-/l* 95.1%

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

      2. *-commutative 95.1%

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

      3. associate-*r/ 95.1%

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

      4. *-commutative 95.1%

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

      5. associate-*l* 99.8%

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

      6. associate-*l* 98.0%

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

   7. Applied egg-rr 98.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 2.7 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(w \cdot \left(r \cdot \left(w \cdot \frac{r}{v + -1}\right)\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \frac{r}{v + -1}\\ \mathbf{if}\;r \leq 4.8 \cdot 10^{-109}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{elif}\;r \leq 6.5 \cdot 10^{+43}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot t\_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot t\_1\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (/ r (+ v -1.0))))
   (if (<= r 4.8e-109)
     (+ t_0 -1.5)
     (if (<= r 6.5e+43)
       (+ t_0 (+ -1.5 (* (* v -0.25) (* r (* (* w w) t_1)))))
       (+
        3.0
        (- (* (* 0.125 (+ 3.0 (* v -2.0))) (* w (* (* r w) t_1))) 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r / (v + -1.0);
	double tmp;
	if (r <= 4.8e-109) {
		tmp = t_0 + -1.5;
	} else if (r <= 6.5e+43) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * t_1))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * ((r * w) * t_1))) - 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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = r / (v + (-1.0d0))
    if (r <= 4.8d-109) then
        tmp = t_0 + (-1.5d0)
    else if (r <= 6.5d+43) then
        tmp = t_0 + ((-1.5d0) + ((v * (-0.25d0)) * (r * ((w * w) * t_1))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * (w * ((r * w) * t_1))) - 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 t_1 = r / (v + -1.0);
	double tmp;
	if (r <= 4.8e-109) {
		tmp = t_0 + -1.5;
	} else if (r <= 6.5e+43) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * t_1))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * ((r * w) * t_1))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = r / (v + -1.0)
	tmp = 0
	if r <= 4.8e-109:
		tmp = t_0 + -1.5
	elif r <= 6.5e+43:
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * t_1))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * ((r * w) * t_1))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(r / Float64(v + -1.0))
	tmp = 0.0
	if (r <= 4.8e-109)
		tmp = Float64(t_0 + -1.5);
	elseif (r <= 6.5e+43)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(v * -0.25) * Float64(r * Float64(Float64(w * w) * t_1)))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(w * Float64(Float64(r * w) * t_1))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = r / (v + -1.0);
	tmp = 0.0;
	if (r <= 4.8e-109)
		tmp = t_0 + -1.5;
	elseif (r <= 6.5e+43)
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * t_1))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * (w * ((r * w) * t_1))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 4.8e-109], N[(t$95$0 + -1.5), $MachinePrecision], If[LessEqual[r, 6.5e+43], N[(t$95$0 + N[(-1.5 + N[(N[(v * -0.25), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(N[(r * w), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 4.79999999999999977e-109

   1. Initial program 79.6%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in v around inf 78.1%

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

      1. *-commutative 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in r around 0 64.9%

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

   if 4.79999999999999977e-109 < r < 6.4999999999999998e43

   1. Initial program 90.9%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in v around inf 88.5%

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

      1. *-commutative 88.5%

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

   7. Simplified 88.5%

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

   if 6.4999999999999998e43 < r

   1. Initial program 93.5%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in r around inf 95.2%

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

      1. associate-/l* 95.3%

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

      2. *-commutative 95.3%

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

      3. associate-*r/ 95.2%

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

      4. associate-*l* 99.8%

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

      5. associate-*r* 99.8%

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

      6. add-sqr-sqrt 99.6%

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

      7. associate-*l* 99.6%

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

      8. add-sqr-sqrt 60.2%

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

      9. sqrt-prod 68.0%

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

      10. sqrt-prod 68.1%

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

      11. sqrt-prod 68.2%

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

      12. associate-*l* 68.2%

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

      13. *-commutative 68.2%

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

      14. associate-*l* 68.2%

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

   7. Applied egg-rr 98.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 6.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{v + -1}\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 4 \cdot 10^{-108}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{elif}\;r \leq 2.25 \cdot 10^{+45}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 4e-108)
     (+ t_0 -1.5)
     (if (<= r 2.25e+45)
       (+ t_0 (+ -1.5 (* (* v -0.25) (* r (* (* w w) (/ r (+ v -1.0)))))))
       (-
        (+ 3.0 (/ (* (+ 0.375 (* v -0.25)) (* r (* r (* w 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 <= 4e-108) {
		tmp = t_0 + -1.5;
	} else if (r <= 2.25e+45) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = (3.0 + (((0.375 + (v * -0.25)) * (r * (r * (w * 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 <= 4d-108) then
        tmp = t_0 + (-1.5d0)
    else if (r <= 2.25d+45) then
        tmp = t_0 + ((-1.5d0) + ((v * (-0.25d0)) * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = (3.0d0 + (((0.375d0 + (v * (-0.25d0))) * (r * (r * (w * 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 <= 4e-108) {
		tmp = t_0 + -1.5;
	} else if (r <= 2.25e+45) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = (3.0 + (((0.375 + (v * -0.25)) * (r * (r * (w * w)))) / (v + -1.0))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 4e-108:
		tmp = t_0 + -1.5
	elif r <= 2.25e+45:
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = (3.0 + (((0.375 + (v * -0.25)) * (r * (r * (w * 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 <= 4e-108)
		tmp = Float64(t_0 + -1.5);
	elseif (r <= 2.25e+45)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(v * -0.25) * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(r * Float64(r * Float64(w * 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 <= 4e-108)
		tmp = t_0 + -1.5;
	elseif (r <= 2.25e+45)
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = (3.0 + (((0.375 + (v * -0.25)) * (r * (r * (w * 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, 4e-108], N[(t$95$0 + -1.5), $MachinePrecision], If[LessEqual[r, 2.25e+45], N[(t$95$0 + N[(-1.5 + N[(N[(v * -0.25), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 4.00000000000000016e-108

   1. Initial program 79.6%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in v around inf 78.1%

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

      1. *-commutative 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in r around 0 64.9%

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

   if 4.00000000000000016e-108 < r < 2.2499999999999999e45

   1. Initial program 90.9%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in v around inf 88.5%

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

      1. *-commutative 88.5%

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

   7. Simplified 88.5%

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

   if 2.2499999999999999e45 < r

   1. Initial program 93.5%

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

   3. Taylor expanded in r around inf 93.5%

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

   4. Taylor expanded in v around 0 93.5%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 2.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 9 \cdot 10^{-112}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{elif}\;r \leq 7 \cdot 10^{+158}:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(0.375 + v \cdot -0.25\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 9e-112)
     (+ t_0 -1.5)
     (if (<= r 7e+158)
       (+ t_0 (- -1.5 (* 0.375 (* r (* r (* w w))))))
       (-
        (+ 3.0 (* (* w (* r (+ 0.375 (* v -0.25)))) (* 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 <= 9e-112) {
		tmp = t_0 + -1.5;
	} else if (r <= 7e+158) {
		tmp = t_0 + (-1.5 - (0.375 * (r * (r * (w * w)))));
	} else {
		tmp = (3.0 + ((w * (r * (0.375 + (v * -0.25)))) * (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 <= 9d-112) then
        tmp = t_0 + (-1.5d0)
    else if (r <= 7d+158) then
        tmp = t_0 + ((-1.5d0) - (0.375d0 * (r * (r * (w * w)))))
    else
        tmp = (3.0d0 + ((w * (r * (0.375d0 + (v * (-0.25d0))))) * (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 <= 9e-112) {
		tmp = t_0 + -1.5;
	} else if (r <= 7e+158) {
		tmp = t_0 + (-1.5 - (0.375 * (r * (r * (w * w)))));
	} else {
		tmp = (3.0 + ((w * (r * (0.375 + (v * -0.25)))) * (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 <= 9e-112:
		tmp = t_0 + -1.5
	elif r <= 7e+158:
		tmp = t_0 + (-1.5 - (0.375 * (r * (r * (w * w)))))
	else:
		tmp = (3.0 + ((w * (r * (0.375 + (v * -0.25)))) * (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 <= 9e-112)
		tmp = Float64(t_0 + -1.5);
	elseif (r <= 7e+158)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(w * Float64(r * Float64(0.375 + Float64(v * -0.25)))) * 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 <= 9e-112)
		tmp = t_0 + -1.5;
	elseif (r <= 7e+158)
		tmp = t_0 + (-1.5 - (0.375 * (r * (r * (w * w)))));
	else
		tmp = (3.0 + ((w * (r * (0.375 + (v * -0.25)))) * (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, 9e-112], N[(t$95$0 + -1.5), $MachinePrecision], If[LessEqual[r, 7e+158], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(w * N[(r * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $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 < 9.00000000000000024e-112

   1. Initial program 79.9%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in v around inf 78.4%

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

      1. *-commutative 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in r around 0 65.1%

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

   if 9.00000000000000024e-112 < r < 7.0000000000000003e158

   1. Initial program 93.1%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in v around 0 81.0%

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

   6. Taylor expanded in v around 0 84.7%

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

   if 7.0000000000000003e158 < r

   1. Initial program 89.1%

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

   3. Taylor expanded in r around inf 89.1%

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

   5. Applied egg-rr 93.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 7 \cdot 10^{+158}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(0.375 + v \cdot -0.25\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 83.9%

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

3. Simplified 85.4%

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

   1. *-commutative 85.4%

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

   2. fma-undefine 85.4%

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

   3. *-commutative 85.4%

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

   4. +-commutative 85.4%

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

   5. metadata-eval 85.4%

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

   6. cancel-sign-sub-inv 85.4%

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

   7. associate-*r/ 85.4%

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

   8. *-commutative 85.4%

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

   9. associate-/l* 85.8%

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

   10. *-commutative 85.8%

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

   11. clear-num 85.8%

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

   12. un-div-inv 85.8%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. unpow2 99.8%

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

   3. times-frac 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 8: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

Python

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

MATLAB

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

Derivation

1. Initial program 83.9%

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

3. Simplified 85.4%

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

   1. *-commutative 85.4%

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

   2. fma-undefine 85.4%

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

   3. *-commutative 85.4%

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

   4. +-commutative 85.4%

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

   5. metadata-eval 85.4%

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

   6. cancel-sign-sub-inv 85.4%

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

   7. associate-*r/ 85.4%

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

   8. *-commutative 85.4%

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

   9. associate-/l* 85.8%

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

   10. *-commutative 85.8%

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

   11. clear-num 85.8%

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

   12. un-div-inv 85.8%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. unpow2 99.8%

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

   3. times-frac 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. *-un-lft-identity 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 9: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) / ((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(v * -0.25)) / 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 + (v * -0.25)) / ((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[(v * -0.25), $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 83.9%

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

3. Simplified 85.4%

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

   1. *-commutative 85.4%

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

   2. fma-undefine 85.4%

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

   3. *-commutative 85.4%

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

   4. +-commutative 85.4%

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

   5. metadata-eval 85.4%

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

   6. cancel-sign-sub-inv 85.4%

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

   7. associate-*r/ 85.4%

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

   8. *-commutative 85.4%

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

   9. associate-/l* 85.8%

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

   10. *-commutative 85.8%

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

   11. clear-num 85.8%

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

   12. un-div-inv 85.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 10: 74.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 10^{-111}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 1e-111)
     (+ t_0 -1.5)
     (+ t_0 (- -1.5 (* 0.375 (* r (* r (* w w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1e-111) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + (-1.5 - (0.375 * (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 <= 1d-111) then
        tmp = t_0 + (-1.5d0)
    else
        tmp = t_0 + ((-1.5d0) - (0.375d0 * (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 <= 1e-111) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + (-1.5 - (0.375 * (r * (r * (w * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 1e-111:
		tmp = t_0 + -1.5
	else:
		tmp = t_0 + (-1.5 - (0.375 * (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 <= 1e-111)
		tmp = Float64(t_0 + -1.5);
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * 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 <= 1e-111)
		tmp = t_0 + -1.5;
	else
		tmp = t_0 + (-1.5 - (0.375 * (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, 1e-111], N[(t$95$0 + -1.5), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.00000000000000009e-111

   1. Initial program 79.9%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in v around inf 78.4%

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

      1. *-commutative 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in r around 0 65.1%

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

   if 1.00000000000000009e-111 < r

   1. Initial program 91.7%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in v around 0 75.0%

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

   6. Taylor expanded in v around 0 84.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 10^{-111}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 85.4%

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

5. Taylor expanded in v around inf 79.8%

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

   1. *-commutative 79.8%

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

7. Simplified 79.8%

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

8. Taylor expanded in r around 0 56.7%

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]
9. Add Preprocessing

Alternative 12: 14.4% 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 83.9%

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

3. Simplified 85.4%

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

5. Taylor expanded in v around inf 79.8%

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

   1. *-commutative 79.8%

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

7. Simplified 79.8%

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

8. Taylor expanded in r around 0 56.7%

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

9. Taylor expanded in r around inf 12.9%

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

Reproduce

herbie shell --seed 2024186 
(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))