Rosa's TurbineBenchmark

Percentage Accurate: 84.2% → 99.2%

Time: 14.4s

Alternatives: 6

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: 84.2% 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: 99.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 97.1%

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

   1. associate-/r* 97.1%

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

   2. *-un-lft-identity 97.1%

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

   3. *-commutative 97.1%

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

   4. times-frac 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-5}:\\ \;\;\;\;-1.5 + \left(2 \cdot {r}^{-2} + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5e-5)
   (+ -1.5 (+ (* 2.0 (pow r -2.0)) (* -0.25 (* w (* r (* r w))))))
   (+
    (/ (/ 2.0 r) r)
    (- -1.5 (/ (fma v -0.25 0.375) (/ (- 1.0 v) (* r (* w (* r w)))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e-5) {
		tmp = -1.5 + ((2.0 * pow(r, -2.0)) + (-0.25 * (w * (r * (r * w)))));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - (fma(v, -0.25, 0.375) / ((1.0 - v) / (r * (w * (r * w))))));
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5e-5)
		tmp = Float64(-1.5 + Float64(Float64(2.0 * (r ^ -2.0)) + Float64(-0.25 * Float64(w * Float64(r * Float64(r * w))))));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(fma(v, -0.25, 0.375) / Float64(Float64(1.0 - v) / Float64(r * Float64(w * Float64(r * w)))))));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5e-5], N[(-1.5 + N[(N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.00000000000000024e-5

   1. Initial program 81.5%

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

   3. Simplified 83.8%

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

      1. expm1-log1p-u 81.0%

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

      2. expm1-udef 81.0%

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

      3. div-inv 81.0%

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

      4. pow2 81.0%

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

      5. pow-flip 81.0%

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

      6. metadata-eval 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. expm1-def 81.0%

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

      2. expm1-log1p 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in v around inf 81.2%

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

       1. *-commutative 81.1%

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

   11. Simplified 81.2%

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

   12. Taylor expanded in r around 0 78.5%

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

       1. *-commutative 78.5%

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

       2. unpow2 78.5%

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

       3. unpow2 78.5%

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

       4. swap-sqr 96.4%

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

       5. unpow2 96.4%

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

   14. Simplified 96.4%

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

       1. unpow2 96.4%

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

       2. associate-*r* 95.4%

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

   16. Applied egg-rr 95.4%

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

   if 5.00000000000000024e-5 < r

   1. Initial program 84.7%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -64000000000 \lor \neg \left(v \leq 2.2 \cdot 10^{-41}\right):\\ \;\;\;\;t_1 + \left(-1.5 - 0.25 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - 0.375 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (/ (/ 2.0 r) r)))
   (if (or (<= v -64000000000.0) (not (<= v 2.2e-41)))
     (+ t_1 (- -1.5 (* 0.25 t_0)))
     (+ t_1 (- -1.5 (* 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 <= -64000000000.0) || !(v <= 2.2e-41)) {
		tmp = t_1 + (-1.5 - (0.25 * t_0));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = (2.0d0 / r) / r
    if ((v <= (-64000000000.0d0)) .or. (.not. (v <= 2.2d-41))) then
        tmp = t_1 + ((-1.5d0) - (0.25d0 * t_0))
    else
        tmp = t_1 + ((-1.5d0) - (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double tmp;
	if ((v <= -64000000000.0) || !(v <= 2.2e-41)) {
		tmp = t_1 + (-1.5 - (0.25 * t_0));
	} else {
		tmp = t_1 + (-1.5 - (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 <= -64000000000.0) or not (v <= 2.2e-41):
		tmp = t_1 + (-1.5 - (0.25 * t_0))
	else:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if ((v <= -64000000000.0) || !(v <= 2.2e-41))
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.25 * t_0)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = (2.0 / r) / r;
	tmp = 0.0;
	if ((v <= -64000000000.0) || ~((v <= 2.2e-41)))
		tmp = t_1 + (-1.5 - (0.25 * t_0));
	else
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[Or[LessEqual[v, -64000000000.0], N[Not[LessEqual[v, 2.2e-41]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 - N[(0.25 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -6.4e10 or 2.2e-41 < v

   1. Initial program 79.6%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in v around inf 81.1%

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

      1. unpow2 81.1%

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

      2. unpow2 81.1%

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

      3. swap-sqr 99.7%

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

      4. unpow2 99.7%

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

   7. Simplified 99.7%

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

      1. unpow2 99.7%

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

   9. Applied egg-rr 99.7%

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

   if -6.4e10 < v < 2.2e-41

   1. Initial program 85.3%

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

   3. Simplified 97.2%

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

      1. associate-/r* 97.2%

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

      2. *-un-lft-identity 97.2%

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

      3. *-commutative 97.2%

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

      4. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. frac-2neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. frac-times 97.2%

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

      4. neg-mul-1 97.2%

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

      5. distribute-neg-frac 97.2%

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

      6. distribute-rgt-neg-in 97.2%

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

   8. Applied egg-rr 97.2%

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

   9. Taylor expanded in v around 0 78.0%

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

       1. *-commutative 78.0%

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

       2. unpow2 78.0%

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

       3. unpow2 78.0%

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

       4. swap-sqr 99.8%

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

       5. unpow2 99.8%

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

   11. Simplified 99.8%

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

       1. unpow2 87.2%

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

   13. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -64000000000 \lor \neg \left(v \leq 2.2 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(w \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -80000000000:\\ \;\;\;\;-1.5 + \left(t_1 + \left(r \cdot -0.25\right) \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_1 + t_0 \cdot \left(r \cdot -0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* w w))) (t_1 (/ 2.0 (* r r))))
   (if (<= v -80000000000.0)
     (+ -1.5 (+ t_1 (* (* r -0.25) t_0)))
     (+ -1.5 (+ t_1 (* t_0 (* r -0.375)))))))

C

double code(double v, double w, double r) {
	double t_0 = r * (w * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -80000000000.0) {
		tmp = -1.5 + (t_1 + ((r * -0.25) * t_0));
	} else {
		tmp = -1.5 + (t_1 + (t_0 * (r * -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) :: t_1
    real(8) :: tmp
    t_0 = r * (w * w)
    t_1 = 2.0d0 / (r * r)
    if (v <= (-80000000000.0d0)) then
        tmp = (-1.5d0) + (t_1 + ((r * (-0.25d0)) * t_0))
    else
        tmp = (-1.5d0) + (t_1 + (t_0 * (r * (-0.375d0))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * (w * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -80000000000.0) {
		tmp = -1.5 + (t_1 + ((r * -0.25) * t_0));
	} else {
		tmp = -1.5 + (t_1 + (t_0 * (r * -0.375)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * (w * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if v <= -80000000000.0:
		tmp = -1.5 + (t_1 + ((r * -0.25) * t_0))
	else:
		tmp = -1.5 + (t_1 + (t_0 * (r * -0.375)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(w * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -80000000000.0)
		tmp = Float64(-1.5 + Float64(t_1 + Float64(Float64(r * -0.25) * t_0)));
	else
		tmp = Float64(-1.5 + Float64(t_1 + Float64(t_0 * Float64(r * -0.375))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * (w * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -80000000000.0)
		tmp = -1.5 + (t_1 + ((r * -0.25) * t_0));
	else
		tmp = -1.5 + (t_1 + (t_0 * (r * -0.375)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -80000000000.0], N[(-1.5 + N[(t$95$1 + N[(N[(r * -0.25), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$1 + N[(t$95$0 * N[(r * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -8e10

   1. Initial program 76.5%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in v around inf 83.5%

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

      1. *-commutative 83.5%

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

   7. Simplified 83.5%

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

   if -8e10 < v

   1. Initial program 84.6%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in v around 0 84.0%

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

      1. *-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -80000000000:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot -0.25\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot -0.375\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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)) + ((r * (w * w)) * (r * (-0.375d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 85.4%

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

5. Taylor expanded in v around 0 81.1%

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

   1. *-commutative 81.1%

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

7. Simplified 81.1%

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

8. Final simplification 81.1%

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

Alternative 6: 93.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 97.1%

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

5. Taylor expanded in v around inf 76.4%

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

   1. unpow2 76.4%

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

   2. unpow2 76.4%

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

   3. swap-sqr 94.2%

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

   4. unpow2 94.2%

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

7. Simplified 94.2%

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

   1. unpow2 94.2%

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

9. Applied egg-rr 94.2%

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

10. Final simplification 94.2%

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

Reproduce

herbie shell --seed 2024024 
(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))