Rosa's TurbineBenchmark

Percentage Accurate: 84.1% → 99.8%

Time: 13.0s

Alternatives: 9

Speedup: 1.2×

• 52.4% 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: 84.1% 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, 0.2× speedup

Math

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

FPCore

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

C

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

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))) / ((((w ** (-1.0d0)) * (1.0d0 / r)) / (r * w)) * (1.0d0 - v))))
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)) / (((Math.pow(w, -1.0) * (1.0 / r)) / (r * w)) * (1.0 - v))));
}

Python

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

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

MATLAB

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

Derivation

1. Initial program 87.8%

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

3. Simplified 89.0%

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

   1. fma-undefine 89.0%

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

   2. *-commutative 89.0%

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

   3. +-commutative 89.0%

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

   4. associate-*r/ 89.4%

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

   5. *-commutative 89.4%

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

   6. associate-/l* 89.1%

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

   7. clear-num 89.1%

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

   8. un-div-inv 89.1%

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

   9. distribute-rgt-in 89.1%

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

   10. metadata-eval 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 89.1%

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

   14. associate-*r* 78.6%

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

   15. pow2 78.6%

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

   16. pow2 78.6%

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

   17. pow-prod-down 99.5%

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

   18. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

   1. frac-2neg 99.5%

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

   2. div-inv 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

   5. div-inv 99.5%

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

   6. pow-flip 99.8%

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

   7. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   4. fma-undefine 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   5. *-commutative 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   6. +-commutative 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   7. associate--r+ 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. *-commutative 99.8%

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

   11. distribute-rgt-neg-in 99.8%

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

   12. *-commutative 99.8%

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

   13. neg-sub0 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{{\left(r \cdot w\right)}^{-2} \cdot \color{blue}{\left(0 - \left(1 - v\right)\right)}}\right) \]

   14. associate--r- 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{{\left(r \cdot w\right)}^{-2} \cdot \color{blue}{\left(\left(0 - 1\right) + v\right)}}\right) \]

   15. metadata-eval 99.8%

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

10. Simplified 99.8%

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

    1. *-commutative 99.8%

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

    2. metadata-eval 99.8%

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

    3. pow-sqr 99.7%

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

    4. unpow-1 99.7%

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

    5. *-commutative 99.7%

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

    6. unpow-1 99.7%

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

    7. *-commutative 99.7%

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

12. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.8%

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

    2. *-lft-identity 99.8%

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

14. Simplified 99.8%

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

    1. inv-pow 99.8%

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

    2. *-commutative 99.8%

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

    3. unpow-prod-down 99.8%

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

    4. inv-pow 99.8%

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

16. Applied egg-rr 99.8%

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

17. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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))) / (((r * w) ** (-2.0d0)) * (1.0d0 - v))))
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)) / (Math.pow((r * w), -2.0) * (1.0 - v))));
}

Python

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

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(r * w) ^ -2.0) * Float64(1.0 - v)))))
end

MATLAB

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

Derivation

1. Initial program 87.8%

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

3. Simplified 89.0%

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

   1. fma-undefine 89.0%

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

   2. *-commutative 89.0%

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

   3. +-commutative 89.0%

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

   4. associate-*r/ 89.4%

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

   5. *-commutative 89.4%

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

   6. associate-/l* 89.1%

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

   7. clear-num 89.1%

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

   8. un-div-inv 89.1%

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

   9. distribute-rgt-in 89.1%

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

   10. metadata-eval 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 89.1%

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

   14. associate-*r* 78.6%

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

   15. pow2 78.6%

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

   16. pow2 78.6%

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

   17. pow-prod-down 99.5%

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

   18. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

   1. frac-2neg 99.5%

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

   2. div-inv 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

   5. div-inv 99.5%

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

   6. pow-flip 99.8%

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

   7. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   4. fma-undefine 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   5. *-commutative 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   6. +-commutative 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   7. associate--r+ 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. *-commutative 99.8%

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

   11. distribute-rgt-neg-in 99.8%

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

   12. *-commutative 99.8%

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

   13. neg-sub0 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{{\left(r \cdot w\right)}^{-2} \cdot \color{blue}{\left(0 - \left(1 - v\right)\right)}}\right) \]

   14. associate--r- 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{{\left(r \cdot w\right)}^{-2} \cdot \color{blue}{\left(\left(0 - 1\right) + v\right)}}\right) \]

   15. metadata-eval 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 20000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{1}{r \cdot w} \cdot \frac{-1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= v 20000000000.0)
   (+
    (/ 2.0 (* r r))
    (+ -1.5 (/ (+ (* v -0.25) 0.375) (* (/ 1.0 (* r w)) (/ -1.0 (* r w))))))
   (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (v <= 20000000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / ((1.0 / (r * w)) * (-1.0 / (r * w)))));
	} else {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (v <= 20000000000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (((v * (-0.25d0)) + 0.375d0) / ((1.0d0 / (r * w)) * ((-1.0d0) / (r * w)))))
    else
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (v <= 20000000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / ((1.0 / (r * w)) * (-1.0 / (r * w)))));
	} else {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if v <= 20000000000.0:
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / ((1.0 / (r * w)) * (-1.0 / (r * w)))))
	else:
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (v <= 20000000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(Float64(1.0 / Float64(r * w)) * Float64(-1.0 / Float64(r * w))))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (v <= 20000000000.0)
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / ((1.0 / (r * w)) * (-1.0 / (r * w)))));
	else
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[v, 20000000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 2e10

   1. Initial program 90.7%

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

   3. Simplified 92.0%

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

      1. fma-undefine 92.0%

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

      2. *-commutative 92.0%

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

      3. +-commutative 92.0%

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

      4. associate-*r/ 92.1%

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

      5. *-commutative 92.1%

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

      6. associate-/l* 91.7%

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

      7. clear-num 91.7%

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

      8. un-div-inv 91.7%

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

      9. distribute-rgt-in 91.7%

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

      10. metadata-eval 91.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      11. *-commutative 91.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      12. associate-*l* 91.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      13. metadata-eval 91.7%

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

      14. associate-*r* 79.2%

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

      15. pow2 79.2%

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

      16. pow2 79.2%

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

      17. pow-prod-down 99.4%

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

      18. *-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. unpow2 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in v around 0 92.9%

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

   if 2e10 < v

   1. Initial program 79.1%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in r around 0 63.1%

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

      1. associate-/r* 63.1%

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

      2. div-inv 63.0%

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

   7. Applied egg-rr 63.0%

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

      1. associate-*r/ 63.1%

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

      2. *-rgt-identity 63.1%

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

   9. Simplified 63.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 20000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{1}{r \cdot w} \cdot \frac{-1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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) - (((v * (-0.25d0)) - (-0.375d0)) / (((1.0d0 / (r * w)) / (r * w)) * (1.0d0 - v))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Simplified 89.0%

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

   1. fma-undefine 89.0%

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

   2. *-commutative 89.0%

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

   3. +-commutative 89.0%

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

   4. associate-*r/ 89.4%

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

   5. *-commutative 89.4%

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

   6. associate-/l* 89.1%

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

   7. clear-num 89.1%

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

   8. un-div-inv 89.1%

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

   9. distribute-rgt-in 89.1%

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

   10. metadata-eval 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 89.1%

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

   14. associate-*r* 78.6%

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

   15. pow2 78.6%

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

   16. pow2 78.6%

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

   17. pow-prod-down 99.5%

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

   18. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

   1. frac-2neg 99.5%

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

   2. div-inv 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

   5. div-inv 99.5%

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

   6. pow-flip 99.8%

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

   7. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   4. fma-undefine 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   5. *-commutative 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   6. +-commutative 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   7. associate--r+ 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\left(1 - v\right) \cdot {\left(w \cdot r\right)}^{-2}}\right) \]

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. *-commutative 99.8%

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

   11. distribute-rgt-neg-in 99.8%

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

   12. *-commutative 99.8%

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

   13. neg-sub0 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{{\left(r \cdot w\right)}^{-2} \cdot \color{blue}{\left(0 - \left(1 - v\right)\right)}}\right) \]

   14. associate--r- 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{{\left(r \cdot w\right)}^{-2} \cdot \color{blue}{\left(\left(0 - 1\right) + v\right)}}\right) \]

   15. metadata-eval 99.8%

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

10. Simplified 99.8%

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

    1. *-commutative 99.8%

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

    2. metadata-eval 99.8%

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

    3. pow-sqr 99.7%

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

    4. unpow-1 99.7%

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

    5. *-commutative 99.7%

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

    6. unpow-1 99.7%

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

    7. *-commutative 99.7%

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

12. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.8%

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

    2. *-lft-identity 99.8%

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

14. Simplified 99.8%

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

15. Final simplification 99.8%

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

Alternative 5: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 87.8%

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

3. Simplified 89.0%

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

   1. fma-undefine 89.0%

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

   2. *-commutative 89.0%

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

   3. +-commutative 89.0%

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

   4. associate-*r/ 89.4%

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

   5. *-commutative 89.4%

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

   6. associate-/l* 89.1%

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

   7. clear-num 89.1%

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

   8. un-div-inv 89.1%

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

   9. distribute-rgt-in 89.1%

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

   10. metadata-eval 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 89.1%

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

   14. associate-*r* 78.6%

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

   15. pow2 78.6%

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

   16. pow2 78.6%

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

   17. pow-prod-down 99.5%

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

   18. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. unpow2 99.5%

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

   3. times-frac 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Simplified 89.0%

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

   1. fma-undefine 89.0%

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

   2. *-commutative 89.0%

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

   3. +-commutative 89.0%

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

   4. associate-*r/ 89.4%

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

   5. *-commutative 89.4%

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

   6. associate-/l* 89.1%

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

   7. clear-num 89.1%

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

   8. un-div-inv 89.1%

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

   9. distribute-rgt-in 89.1%

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

   10. metadata-eval 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 89.1%

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

   14. associate-*r* 78.6%

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

   15. pow2 78.6%

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

   16. pow2 78.6%

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

   17. pow-prod-down 99.5%

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

   18. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

   1. unpow2 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 7: 57.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Simplified 77.3%

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

5. Taylor expanded in r around 0 60.0%

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

   1. associate-/r* 60.0%

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

   2. div-inv 60.0%

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

7. Applied egg-rr 60.0%

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

   1. associate-*r/ 60.0%

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

   2. *-rgt-identity 60.0%

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

9. Simplified 60.0%

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

Alternative 8: 57.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Simplified 77.3%

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

5. Taylor expanded in r around 0 60.0%

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

6. Final simplification 60.0%

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

Alternative 9: 14.3% 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 87.8%

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

3. Simplified 77.3%

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

5. Taylor expanded in r around 0 60.0%

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

6. Taylor expanded in r around inf 18.5%

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

Reproduce

herbie shell --seed 2024103 
(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))