Rosa's TurbineBenchmark

Percentage Accurate: 84.6% → 99.8%

Time: 11.8s

Alternatives: 7

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: 84.6% 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

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

FPCore

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

C

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

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)) + (((v * (-0.25d0)) - (-0.375d0)) / ((v + (-1.0d0)) * ((r * w) ** (-2.0d0))))
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) * Math.pow((r * w), -2.0)));
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Simplified 95.7%

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

   1. *-commutative 95.7%

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

   2. clear-num 95.7%

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

   3. *-commutative 95.7%

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

   4. associate-*r* 85.3%

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

   5. un-div-inv 85.3%

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

   6. fma-udef 85.3%

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

   7. metadata-eval 85.3%

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

   8. associate-*l* 85.2%

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

   9. *-commutative 85.2%

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

   10. metadata-eval 85.2%

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

   11. distribute-rgt-in 85.2%

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

   12. +-commutative 85.2%

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

   13. clear-num 85.2%

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

5. Applied egg-rr 99.8%

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

   1. frac-2neg 99.8%

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

   2. div-inv 99.8%

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

   3. div-inv 99.8%

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

   4. distribute-lft-neg-in 99.8%

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

   5. pow-flip 99.8%

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

   6. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. distribute-neg-frac 99.8%

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

   5. associate-/l* 95.5%

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

   6. distribute-neg-frac 95.5%

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

   7. neg-sub0 95.5%

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

   8. associate--r- 95.5%

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

   9. metadata-eval 95.5%

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

   10. neg-sub0 95.5%

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

   11. fma-udef 95.5%

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

   12. *-commutative 95.5%

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

   13. +-commutative 95.5%

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

   14. *-commutative 95.5%

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

   15. associate--r+ 95.5%

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

   16. metadata-eval 95.5%

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

9. Simplified 95.5%

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

    1. expm1-log1p-u 94.3%

       \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{-1 + v}{\frac{-0.375 - v \cdot -0.25}{{\left(r \cdot w\right)}^{-2}}}}\right)\right)} \]

    2. expm1-udef 94.3%

       \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{-1 + v}{\frac{-0.375 - v \cdot -0.25}{{\left(r \cdot w\right)}^{-2}}}}\right)} - 1\right)} \]

    3. associate-/r/ 94.3%

       \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{-1 + v} \cdot \frac{-0.375 - v \cdot -0.25}{{\left(r \cdot w\right)}^{-2}}}\right)} - 1\right) \]

    4. frac-times 98.3%

       \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(-0.375 - v \cdot -0.25\right)}{\left(-1 + v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right)} - 1\right) \]

    5. *-un-lft-identity 98.3%

       \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{-0.375 - v \cdot -0.25}}{\left(-1 + v\right) \cdot {\left(r \cdot w\right)}^{-2}}\right)} - 1\right) \]

11. Applied egg-rr 98.3%

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.375 - v \cdot -0.25}{\left(-1 + v\right) \cdot {\left(r \cdot w\right)}^{-2}}\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 98.3%

       \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.375 - v \cdot -0.25}{\left(-1 + v\right) \cdot {\left(r \cdot w\right)}^{-2}}\right)\right)} \]

    2. expm1-log1p 99.8%

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

    3. +-commutative 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (3.0 + ((2.0 / (r * r)) - ((0.125 * (3.0 + (v * -2.0))) / ((1.0 / (r * w)) * ((1.0 - v) / (r * w)))))) + -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 + (v * (-2.0d0)))) / ((1.0d0 / (r * w)) * ((1.0d0 - v) / (r * w)))))) + (-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 + (v * -2.0))) / ((1.0 / (r * w)) * ((1.0 - v) / (r * w)))))) + -4.5;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Simplified 85.2%

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

   1. associate-*r* 95.3%

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

   2. *-commutative 95.3%

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

   3. *-un-lft-identity 95.3%

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

   4. associate-*r* 99.4%

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

   5. times-frac 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 220:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \frac{r \cdot \left(w \cdot \left(v \cdot -0.25 + 0.375\right)\right)}{1 - v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 220.0)
   (+ -1.5 (+ (/ (/ 2.0 r) r) (* -0.375 (* (* r w) (* r w)))))
   (+
    -4.5
    (+
     3.0
     (-
      (/ 2.0 (* r r))
      (* (* r w) (/ (* r (* w (+ (* v -0.25) 0.375))) (- 1.0 v))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 220.0) {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.375 * ((r * w) * (r * w))));
	} else {
		tmp = -4.5 + (3.0 + ((2.0 / (r * r)) - ((r * w) * ((r * (w * ((v * -0.25) + 0.375))) / (1.0 - v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 220.0d0) then
        tmp = (-1.5d0) + (((2.0d0 / r) / r) + ((-0.375d0) * ((r * w) * (r * w))))
    else
        tmp = (-4.5d0) + (3.0d0 + ((2.0d0 / (r * r)) - ((r * w) * ((r * (w * ((v * (-0.25d0)) + 0.375d0))) / (1.0d0 - v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 220.0) {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.375 * ((r * w) * (r * w))));
	} else {
		tmp = -4.5 + (3.0 + ((2.0 / (r * r)) - ((r * w) * ((r * (w * ((v * -0.25) + 0.375))) / (1.0 - v)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 220.0:
		tmp = -1.5 + (((2.0 / r) / r) + (-0.375 * ((r * w) * (r * w))))
	else:
		tmp = -4.5 + (3.0 + ((2.0 / (r * r)) - ((r * w) * ((r * (w * ((v * -0.25) + 0.375))) / (1.0 - v)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 220.0)
		tmp = Float64(-1.5 + Float64(Float64(Float64(2.0 / r) / r) + Float64(-0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(Float64(2.0 / Float64(r * r)) - Float64(Float64(r * w) * Float64(Float64(r * Float64(w * Float64(Float64(v * -0.25) + 0.375))) / Float64(1.0 - v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 220.0)
		tmp = -1.5 + (((2.0 / r) / r) + (-0.375 * ((r * w) * (r * w))));
	else
		tmp = -4.5 + (3.0 + ((2.0 / (r * r)) - ((r * w) * ((r * (w * ((v * -0.25) + 0.375))) / (1.0 - v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 220.0], N[(-1.5 + N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(N[(r * N[(w * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 220

   1. Initial program 82.6%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in v around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. unpow2 77.0%

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

      4. swap-sqr 96.6%

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

      5. unpow2 96.6%

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

   6. Simplified 96.6%

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

      1. unpow2 96.6%

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

   8. Applied egg-rr 96.6%

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

      1. expm1-log1p-u 95.0%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)\right)} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

      2. expm1-udef 95.0%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)} - 1\right)} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

      3. div-inv 95.0%

         \[\leadsto \left(\left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{r \cdot r}}\right)} - 1\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

      4. pow2 95.0%

         \[\leadsto \left(\left(e^{\mathsf{log1p}\left(2 \cdot \frac{1}{\color{blue}{{r}^{2}}}\right)} - 1\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

      5. pow-flip 95.0%

         \[\leadsto \left(\left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right)} - 1\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

      6. metadata-eval 95.0%

         \[\leadsto \left(\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{\color{blue}{-2}}\right)} - 1\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

   10. Applied egg-rr 95.0%

       \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)} - 1\right)} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]
   11. Step-by-step derivation

       1. expm1-def 95.0%

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)\right)} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

       2. expm1-log1p 96.7%

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

   12. Simplified 96.7%

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

       1. metadata-eval 96.7%

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

       2. pow-flip 96.6%

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

       3. pow2 96.6%

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

       4. div-inv 96.6%

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

       5. associate-/r* 96.6%

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

   14. Applied egg-rr 96.6%

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

   if 220 < r

   1. Initial program 83.1%

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

   3. Simplified 89.3%

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

      1. associate-/r/ 89.3%

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

      2. associate-*r* 77.6%

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

      3. swap-sqr 98.1%

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

      4. associate-*r* 98.1%

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

      5. +-commutative 98.1%

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

      6. distribute-rgt-in 98.1%

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

      7. *-commutative 98.1%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.8%

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

      11. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in r around 0 93.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 220:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \frac{r \cdot \left(w \cdot \left(v \cdot -0.25 + 0.375\right)\right)}{1 - v}\right)\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -300000000000 \lor \neg \left(v \leq 3.1 \cdot 10^{-35}\right):\\ \;\;\;\;\left(t_0 + -1.5\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -300000000000.0) (not (<= v 3.1e-35)))
     (- (+ t_0 -1.5) (* (* (* r w) (* r w)) 0.25))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* r (* w 0.375)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -300000000000.0) || !(v <= 3.1e-35)) {
		tmp = (t_0 + -1.5) - (((r * w) * (r * w)) * 0.25);
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r * (w * 0.375)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-300000000000.0d0)) .or. (.not. (v <= 3.1d-35))) then
        tmp = (t_0 + (-1.5d0)) - (((r * w) * (r * w)) * 0.25d0)
    else
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (r * (w * 0.375d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -300000000000.0) || !(v <= 3.1e-35)) {
		tmp = (t_0 + -1.5) - (((r * w) * (r * w)) * 0.25);
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r * (w * 0.375)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -300000000000.0) or not (v <= 3.1e-35):
		tmp = (t_0 + -1.5) - (((r * w) * (r * w)) * 0.25)
	else:
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r * (w * 0.375)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -300000000000.0) || !(v <= 3.1e-35))
		tmp = Float64(Float64(t_0 + -1.5) - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(r * Float64(w * 0.375))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -300000000000.0) || ~((v <= 3.1e-35)))
		tmp = (t_0 + -1.5) - (((r * w) * (r * w)) * 0.25);
	else
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r * (w * 0.375)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -300000000000.0], N[Not[LessEqual[v, 3.1e-35]], $MachinePrecision]], N[(N[(t$95$0 + -1.5), $MachinePrecision] - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -3e11 or 3.10000000000000012e-35 < v

   1. Initial program 78.9%

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

   3. Simplified 96.6%

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

      1. *-commutative 96.6%

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

      2. clear-num 96.6%

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

      3. *-commutative 96.6%

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

      4. associate-*r* 84.1%

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

      5. un-div-inv 84.1%

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

      6. fma-udef 84.1%

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

      7. metadata-eval 84.1%

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

      8. associate-*l* 84.0%

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

      9. *-commutative 84.0%

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

      10. metadata-eval 84.0%

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

      11. distribute-rgt-in 84.0%

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

      12. +-commutative 84.0%

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

      13. clear-num 84.0%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 78.2%

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

      1. *-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. unpow2 78.2%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   8. Simplified 99.8%

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

      1. unpow2 88.9%

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

   10. Applied egg-rr 99.8%

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

   if -3e11 < v < 3.10000000000000012e-35

   1. Initial program 86.4%

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

   3. Simplified 86.4%

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

      1. associate-/r/ 86.4%

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

      2. associate-*r* 78.6%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.8%

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

      11. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -300000000000 \lor \neg \left(v \leq 3.1 \cdot 10^{-35}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -300000000000 \lor \neg \left(v \leq 3.1 \cdot 10^{-35}\right):\\ \;\;\;\;\left(t_1 + -1.5\right) - t_0 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_1 + -0.375 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -300000000000.0) (not (<= v 3.1e-35)))
     (- (+ t_1 -1.5) (* t_0 0.25))
     (+ -1.5 (+ t_1 (* -0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -300000000000.0) || !(v <= 3.1e-35)) {
		tmp = (t_1 + -1.5) - (t_0 * 0.25);
	} else {
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-300000000000.0d0)) .or. (.not. (v <= 3.1d-35))) then
        tmp = (t_1 + (-1.5d0)) - (t_0 * 0.25d0)
    else
        tmp = (-1.5d0) + (t_1 + ((-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 t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -300000000000.0) || !(v <= 3.1e-35)) {
		tmp = (t_1 + -1.5) - (t_0 * 0.25);
	} else {
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -300000000000.0) or not (v <= 3.1e-35):
		tmp = (t_1 + -1.5) - (t_0 * 0.25)
	else:
		tmp = -1.5 + (t_1 + (-0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -300000000000.0) || !(v <= 3.1e-35))
		tmp = Float64(Float64(t_1 + -1.5) - Float64(t_0 * 0.25));
	else
		tmp = Float64(-1.5 + Float64(t_1 + Float64(-0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -300000000000.0) || ~((v <= 3.1e-35)))
		tmp = (t_1 + -1.5) - (t_0 * 0.25);
	else
		tmp = -1.5 + (t_1 + (-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]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -300000000000.0], N[Not[LessEqual[v, 3.1e-35]], $MachinePrecision]], N[(N[(t$95$1 + -1.5), $MachinePrecision] - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$1 + N[(-0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3e11 or 3.10000000000000012e-35 < v

   1. Initial program 78.9%

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

   3. Simplified 96.6%

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

      1. *-commutative 96.6%

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

      2. clear-num 96.6%

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

      3. *-commutative 96.6%

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

      4. associate-*r* 84.1%

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

      5. un-div-inv 84.1%

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

      6. fma-udef 84.1%

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

      7. metadata-eval 84.1%

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

      8. associate-*l* 84.0%

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

      9. *-commutative 84.0%

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

      10. metadata-eval 84.0%

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

      11. distribute-rgt-in 84.0%

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

      12. +-commutative 84.0%

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

      13. clear-num 84.0%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 78.2%

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

      1. *-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. unpow2 78.2%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   8. Simplified 99.8%

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

      1. unpow2 88.9%

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

   10. Applied egg-rr 99.8%

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

   if -3e11 < v < 3.10000000000000012e-35

   1. Initial program 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in v around 0 78.6%

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

      1. *-commutative 78.6%

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

      2. unpow2 78.6%

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

      3. unpow2 78.6%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -300000000000 \lor \neg \left(v \leq 3.1 \cdot 10^{-35}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 6: 93.3% 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 82.7%

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

3. Simplified 85.3%

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

4. Taylor expanded in v around 0 75.9%

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

   1. *-commutative 75.9%

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

   2. unpow2 75.9%

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

   3. unpow2 75.9%

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

   4. swap-sqr 94.5%

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

   5. unpow2 94.5%

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

6. Simplified 94.5%

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

   1. unpow2 94.5%

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

8. Applied egg-rr 94.5%

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

9. Final simplification 94.5%

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

Alternative 7: 93.3% accurate, 1.7× speedup

Math

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

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(Float64(2.0 / 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[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

3. Simplified 85.3%

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

4. Taylor expanded in v around 0 75.9%

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

   1. *-commutative 75.9%

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

   2. unpow2 75.9%

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

   3. unpow2 75.9%

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

   4. swap-sqr 94.5%

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

   5. unpow2 94.5%

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

6. Simplified 94.5%

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

   1. unpow2 94.5%

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

8. Applied egg-rr 94.5%

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

   1. expm1-log1p-u 93.3%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)\right)} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

   2. expm1-udef 93.3%

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)} - 1\right)} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

   3. div-inv 93.3%

      \[\leadsto \left(\left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{r \cdot r}}\right)} - 1\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

   4. pow2 93.3%

      \[\leadsto \left(\left(e^{\mathsf{log1p}\left(2 \cdot \frac{1}{\color{blue}{{r}^{2}}}\right)} - 1\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

   5. pow-flip 93.3%

      \[\leadsto \left(\left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right)} - 1\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

   6. metadata-eval 93.3%

      \[\leadsto \left(\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{\color{blue}{-2}}\right)} - 1\right) + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

10. Applied egg-rr 93.3%

    \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)} - 1\right)} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]
11. Step-by-step derivation

    1. expm1-def 93.3%

       \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)\right)} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5 \]

    2. expm1-log1p 94.6%

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

12. Simplified 94.6%

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

    1. metadata-eval 94.6%

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

    2. pow-flip 94.5%

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

    3. pow2 94.5%

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

    4. div-inv 94.5%

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

    5. associate-/r* 94.5%

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

14. Applied egg-rr 94.5%

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

15. Final simplification 94.5%

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

Reproduce

herbie shell --seed 2023307 
(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))