Rosa's TurbineBenchmark

Percentage Accurate: 85.2% → 99.8%

Time: 19.9s

Alternatives: 5

Speedup: 1.7×

• 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: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 95.5%

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

   1. frac-2neg 95.5%

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

   2. *-commutative 95.5%

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

   3. associate-*r* 85.7%

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

   4. div-inv 85.7%

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

   5. associate-*r* 95.5%

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

   6. *-commutative 95.5%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

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

   4. fma-udef 99.8%

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

   5. *-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate--r+ 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

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

   12. associate--r- 99.8%

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

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. div-inv 99.8%

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

   2. div-inv 99.8%

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

   3. pow-flip 99.8%

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

   4. metadata-eval 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-/r* 99.8%

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

12. Simplified 99.8%

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

    1. *-un-lft-identity 99.8%

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

    2. add-sqr-sqrt 99.7%

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

    3. times-frac 99.7%

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

    4. sqrt-pow1 74.0%

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

    5. metadata-eval 74.0%

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

    6. unpow-1 74.0%

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

    7. sqrt-pow1 99.7%

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

    8. metadata-eval 99.7%

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

    9. unpow-1 99.7%

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

14. Applied egg-rr 99.7%

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

    1. remove-double-div 99.7%

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

    2. associate-*r/ 99.8%

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

16. Simplified 99.8%

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

    1. inv-pow 99.8%

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

    2. unpow-prod-down 99.8%

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

    3. inv-pow 99.8%

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

18. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.8%

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

    2. *-lft-identity 99.8%

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

    3. unpow-1 99.8%

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

20. Simplified 99.8%

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

21. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -3.4 \cdot 10^{+25} \lor \neg \left(v \leq 3200000\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - t_0 \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))))
   (if (or (<= v -3.4e+25) (not (<= v 3200000.0)))
     (+ (/ (/ 2.0 r) r) (- -1.5 (* t_0 0.25)))
     (+ -1.5 (+ (/ 2.0 (* r r)) (* -0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if ((v <= -3.4e+25) || !(v <= 3200000.0)) {
		tmp = ((2.0 / r) / r) + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    if ((v <= (-3.4d+25)) .or. (.not. (v <= 3200000.0d0))) then
        tmp = ((2.0d0 / r) / r) + ((-1.5d0) - (t_0 * 0.25d0))
    else
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((-0.375d0) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if ((v <= -3.4e+25) || !(v <= 3200000.0)) {
		tmp = ((2.0 / r) / r) + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	tmp = 0
	if (v <= -3.4e+25) or not (v <= 3200000.0):
		tmp = ((2.0 / r) / r) + (-1.5 - (t_0 * 0.25))
	else:
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	tmp = 0.0
	if ((v <= -3.4e+25) || !(v <= 3200000.0))
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(t_0 * 0.25)));
	else
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(-0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	tmp = 0.0;
	if ((v <= -3.4e+25) || ~((v <= 3200000.0)))
		tmp = ((2.0 / r) / r) + (-1.5 - (t_0 * 0.25));
	else
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -3.4e+25], N[Not[LessEqual[v, 3200000.0]], $MachinePrecision]], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.39999999999999984e25 or 3.2e6 < v

   1. Initial program 78.2%

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

   3. Simplified 94.8%

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

      1. frac-2neg 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-*r* 81.5%

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

      4. div-inv 81.5%

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

      5. associate-*r* 94.7%

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

      6. *-commutative 94.7%

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

      7. associate-*r* 99.8%

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

      8. pow2 99.8%

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

      9. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. neg-sub0 99.9%

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

      4. fma-udef 99.9%

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

      5. *-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. associate--r+ 99.9%

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

      8. metadata-eval 99.9%

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

      9. *-commutative 99.9%

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

      10. distribute-neg-frac 99.9%

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

      11. neg-sub0 99.9%

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

      12. associate--r- 99.9%

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

      13. metadata-eval 99.9%

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

      14. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in v around inf 76.7%

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

       1. *-commutative 76.7%

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

       2. unpow2 76.7%

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

       3. unpow2 76.7%

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

       4. swap-sqr 99.2%

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

       5. unpow2 99.2%

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

   11. Simplified 99.2%

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

       1. unpow2 86.9%

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

   13. Applied egg-rr 99.2%

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

   if -3.39999999999999984e25 < v < 3.2e6

   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 80.2%

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

   5. Taylor expanded in v around 0 80.2%

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

      1. *-commutative 80.2%

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

      2. unpow2 80.2%

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

      3. unpow2 80.2%

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

      4. swap-sqr 99.7%

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

      5. unpow2 99.7%

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

      6. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. unpow2 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.4 \cdot 10^{+25} \lor \neg \left(v \leq 3200000\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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) * ((r * w) * ((-1.0d0) / (v + (-1.0d0)))))))
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)) * ((r * w) * ((r * w) * (-1.0 / (v + -1.0))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 95.5%

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

   1. frac-2neg 95.5%

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

   2. *-commutative 95.5%

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

   3. associate-*r* 85.7%

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

   4. div-inv 85.7%

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

   5. associate-*r* 95.5%

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

   6. *-commutative 95.5%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

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

   4. fma-udef 99.8%

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

   5. *-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate--r+ 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

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

   12. associate--r- 99.8%

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

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. div-inv 99.8%

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

   2. div-inv 99.8%

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

   3. pow-flip 99.8%

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

   4. metadata-eval 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-/r* 99.8%

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

12. Simplified 99.8%

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

    1. *-un-lft-identity 99.8%

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

    2. add-sqr-sqrt 99.7%

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

    3. times-frac 99.7%

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

    4. sqrt-pow1 74.0%

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

    5. metadata-eval 74.0%

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

    6. unpow-1 74.0%

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

    7. sqrt-pow1 99.7%

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

    8. metadata-eval 99.7%

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

    9. unpow-1 99.7%

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

14. Applied egg-rr 99.7%

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

    1. remove-double-div 99.7%

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

    2. associate-/r/ 99.8%

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

    3. /-rgt-identity 99.8%

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

16. Simplified 99.8%

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

17. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 95.5%

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

   1. frac-2neg 95.5%

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

   2. *-commutative 95.5%

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

   3. associate-*r* 85.7%

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

   4. div-inv 85.7%

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

   5. associate-*r* 95.5%

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

   6. *-commutative 95.5%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

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

   4. fma-udef 99.8%

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

   5. *-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate--r+ 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

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

   12. associate--r- 99.8%

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

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. unpow2 93.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 5: 93.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 77.3%

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

5. Taylor expanded in v around 0 76.3%

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

   1. *-commutative 76.3%

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

   2. unpow2 76.3%

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

   3. unpow2 76.3%

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

   4. swap-sqr 93.8%

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

   5. unpow2 93.8%

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

   6. *-commutative 93.8%

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

7. Simplified 93.8%

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

   1. unpow2 93.8%

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

9. Applied egg-rr 93.8%

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

10. Final simplification 93.8%

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

Reproduce

herbie shell --seed 2024023 
(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))