Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.8%

Time: 12.0s

Alternatives: 9

Speedup: 1.2×

• 53.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

   1. associate--l- 85.3%

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

   2. associate-*l* 78.7%

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

   3. sqr-neg 78.7%

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

   4. associate-*l* 85.3%

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

   5. associate-/l* 87.4%

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

   6. fma-define 87.5%

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

3. Simplified 87.4%

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

   1. *-un-lft-identity 87.4%

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

   2. add-sqr-sqrt 87.4%

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

   3. times-frac 87.4%

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

   4. associate-*r* 80.9%

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

   5. sqrt-prod 80.9%

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

   6. sqrt-prod 41.9%

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

   7. add-sqr-sqrt 72.5%

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

   8. sqrt-prod 38.4%

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

   9. add-sqr-sqrt 75.6%

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

   10. associate-*r* 65.8%

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

   11. sqrt-prod 65.8%

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

   12. sqrt-prod 37.5%

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

   13. add-sqr-sqrt 75.6%

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

   14. sqrt-prod 50.2%

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

   15. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

   1. div-inv 99.8%

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

   2. pow2 99.8%

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

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -0.0016 \lor \neg \left(v \leq 0.000165\right):\\ \;\;\;\;t\_0 + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right) \cdot 0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -0.0016) (not (<= v 0.000165)))
     (+
      t_0
      (+ -1.5 (* (+ 0.375 (* v -0.25)) (/ (* r (* r (* w w))) (+ v -1.0)))))
     (+ t_0 (- -1.5 (* (* (* r w) (/ (* r w) (- 1.0 v))) 0.375))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -0.0016) || !(v <= 0.000165)) {
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 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 <= (-0.0016d0)) .or. (.not. (v <= 0.000165d0))) then
        tmp = t_0 + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) * ((r * (r * (w * w))) / (v + (-1.0d0)))))
    else
        tmp = t_0 + ((-1.5d0) - (((r * w) * ((r * w) / (1.0d0 - v))) * 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 <= -0.0016) || !(v <= 0.000165)) {
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -0.0016) or not (v <= 0.000165):
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0))))
	else:
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 0.375))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -0.0016) || !(v <= 0.000165))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(v + -1.0)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))) * 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 <= -0.0016) || ~((v <= 0.000165)))
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0))));
	else
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 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, -0.0016], N[Not[LessEqual[v, 0.000165]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -0.00160000000000000008 or 1.65e-4 < v

   1. Initial program 88.6%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in v around 0 93.0%

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

      1. *-commutative 93.0%

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

   7. Simplified 93.0%

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

   if -0.00160000000000000008 < v < 1.65e-4

   1. Initial program 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in v around 0 82.0%

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

      1. add-sqr-sqrt 82.0%

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

      2. *-un-lft-identity 82.0%

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

      3. times-frac 82.0%

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

      4. associate-*r* 73.5%

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

      5. sqrt-prod 73.5%

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

      6. sqrt-prod 35.2%

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

      7. add-sqr-sqrt 70.3%

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

      8. sqrt-prod 41.3%

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

      9. add-sqr-sqrt 78.7%

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

      10. associate-*r* 64.7%

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

      11. sqrt-prod 64.7%

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

      12. sqrt-prod 35.7%

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

      13. add-sqr-sqrt 74.7%

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

      14. sqrt-prod 51.8%

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

      15. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -0.0016 \lor \neg \left(v \leq 0.000165\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right) \cdot 0.375\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\\ \mathbf{if}\;r \leq 3.5 \cdot 10^{-91}:\\ \;\;\;\;t\_0 + \left(-1.5 - t\_1 \cdot 0.375\right)\\ \mathbf{elif}\;r \leq 10^{+196}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot t\_1 + 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* (* r w) (/ (* r w) (- 1.0 v)))))
   (if (<= r 3.5e-91)
     (+ t_0 (- -1.5 (* t_1 0.375)))
     (if (<= r 1e+196)
       (+
        t_0
        (+ -1.5 (* (+ 0.375 (* v -0.25)) (/ (* r (* r (* w w))) (+ v -1.0)))))
       (- 3.0 (+ (* (* 0.125 (+ 3.0 (* -2.0 v))) t_1) 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (r * w) * ((r * w) / (1.0 - v));
	double tmp;
	if (r <= 3.5e-91) {
		tmp = t_0 + (-1.5 - (t_1 * 0.375));
	} else if (r <= 1e+196) {
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = 3.0 - (((0.125 * (3.0 + (-2.0 * v))) * t_1) + 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (r * w) * ((r * w) / (1.0d0 - v))
    if (r <= 3.5d-91) then
        tmp = t_0 + ((-1.5d0) - (t_1 * 0.375d0))
    else if (r <= 1d+196) then
        tmp = t_0 + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) * ((r * (r * (w * w))) / (v + (-1.0d0)))))
    else
        tmp = 3.0d0 - (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * t_1) + 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (r * w) * ((r * w) / (1.0 - v));
	double tmp;
	if (r <= 3.5e-91) {
		tmp = t_0 + (-1.5 - (t_1 * 0.375));
	} else if (r <= 1e+196) {
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = 3.0 - (((0.125 * (3.0 + (-2.0 * v))) * t_1) + 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (r * w) * ((r * w) / (1.0 - v))
	tmp = 0
	if r <= 3.5e-91:
		tmp = t_0 + (-1.5 - (t_1 * 0.375))
	elif r <= 1e+196:
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0))))
	else:
		tmp = 3.0 - (((0.125 * (3.0 + (-2.0 * v))) * t_1) + 4.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v)))
	tmp = 0.0
	if (r <= 3.5e-91)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * 0.375)));
	elseif (r <= 1e+196)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(v + -1.0)))));
	else
		tmp = Float64(3.0 - Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * t_1) + 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (r * w) * ((r * w) / (1.0 - v));
	tmp = 0.0;
	if (r <= 3.5e-91)
		tmp = t_0 + (-1.5 - (t_1 * 0.375));
	elseif (r <= 1e+196)
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0))));
	else
		tmp = 3.0 - (((0.125 * (3.0 + (-2.0 * v))) * t_1) + 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 3.5e-91], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1e+196], N[(t$95$0 + N[(-1.5 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 3.4999999999999999e-91

   1. Initial program 84.2%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in v around 0 79.6%

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

      1. add-sqr-sqrt 84.2%

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

      2. *-un-lft-identity 84.2%

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

      3. times-frac 84.2%

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

      4. associate-*r* 79.4%

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

      5. sqrt-prod 79.3%

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

      6. sqrt-prod 21.1%

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

      7. add-sqr-sqrt 63.5%

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

      8. sqrt-prod 33.1%

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

      9. add-sqr-sqrt 78.7%

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

      10. associate-*r* 68.9%

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

      11. sqrt-prod 68.9%

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

      12. sqrt-prod 25.9%

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

      13. add-sqr-sqrt 78.6%

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

      14. sqrt-prod 48.5%

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

      15. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 89.7%

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

   if 3.4999999999999999e-91 < r < 9.9999999999999995e195

   1. Initial program 89.1%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in v around 0 98.5%

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

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   if 9.9999999999999995e195 < r

   1. Initial program 82.1%

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

      1. associate--l- 82.1%

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

      2. associate-*l* 55.7%

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

      3. sqr-neg 55.7%

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

      4. associate-*l* 82.1%

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

      5. associate-/l* 82.1%

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

      6. fma-define 82.1%

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

   3. Simplified 82.1%

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

      1. *-un-lft-identity 82.1%

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

      2. add-sqr-sqrt 82.2%

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

      3. times-frac 82.2%

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

      4. associate-*r* 55.7%

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

      5. sqrt-prod 55.7%

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

      6. sqrt-prod 82.2%

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

      7. add-sqr-sqrt 82.2%

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

      8. sqrt-prod 73.4%

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

      9. add-sqr-sqrt 73.4%

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

      10. associate-*r* 55.2%

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

      11. sqrt-prod 55.2%

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

      12. sqrt-prod 73.4%

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

      13. add-sqr-sqrt 73.4%

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

      14. sqrt-prod 90.6%

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

      15. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around inf 99.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right) \cdot 0.375\right)\\ \mathbf{elif}\;r \leq 10^{+196}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right) + 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -21 \lor \neg \left(v \leq 1.6\right):\\ \;\;\;\;t\_0 + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right) \cdot 0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -21.0) (not (<= v 1.6)))
     (+ t_0 (+ -1.5 (* (* v -0.25) (/ (* r (* r (* w w))) (+ v -1.0)))))
     (+ t_0 (- -1.5 (* (* (* r w) (/ (* r w) (- 1.0 v))) 0.375))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -21.0) || !(v <= 1.6)) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 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 <= (-21.0d0)) .or. (.not. (v <= 1.6d0))) then
        tmp = t_0 + ((-1.5d0) + ((v * (-0.25d0)) * ((r * (r * (w * w))) / (v + (-1.0d0)))))
    else
        tmp = t_0 + ((-1.5d0) - (((r * w) * ((r * w) / (1.0d0 - v))) * 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 <= -21.0) || !(v <= 1.6)) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -21.0) or not (v <= 1.6):
		tmp = t_0 + (-1.5 + ((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))))
	else:
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 0.375))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -21.0) || !(v <= 1.6))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(v * -0.25) * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(v + -1.0)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))) * 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 <= -21.0) || ~((v <= 1.6)))
		tmp = t_0 + (-1.5 + ((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))));
	else
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 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, -21.0], N[Not[LessEqual[v, 1.6]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -21 or 1.6000000000000001 < v

   1. Initial program 88.3%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in v around inf 92.1%

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

      1. *-commutative 92.1%

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

   7. Simplified 92.1%

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

   if -21 < v < 1.6000000000000001

   1. Initial program 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in v around 0 82.0%

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

      1. add-sqr-sqrt 82.5%

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

      2. *-un-lft-identity 82.5%

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

      3. times-frac 82.5%

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

      4. associate-*r* 73.6%

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

      5. sqrt-prod 73.6%

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

      6. sqrt-prod 36.3%

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

      7. add-sqr-sqrt 70.4%

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

      8. sqrt-prod 41.6%

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

      9. add-sqr-sqrt 79.3%

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

      10. associate-*r* 65.0%

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

      11. sqrt-prod 65.0%

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

      12. sqrt-prod 36.9%

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

      13. add-sqr-sqrt 74.7%

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

      14. sqrt-prod 51.8%

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

      15. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -21 \lor \neg \left(v \leq 1.6\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right) \cdot 0.375\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.5 \cdot 10^{+139}:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.375 \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right) \cdot 0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -1.5e+139)
     (+ t_0 (- -1.5 (* 0.375 (/ (* r (* r (* w w))) (- 1.0 v)))))
     (+ t_0 (- -1.5 (* (* (* r w) (/ (* r w) (- 1.0 v))) 0.375))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -1.5e+139) {
		tmp = t_0 + (-1.5 - (0.375 * ((r * (r * (w * w))) / (1.0 - v))));
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 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 <= (-1.5d+139)) then
        tmp = t_0 + ((-1.5d0) - (0.375d0 * ((r * (r * (w * w))) / (1.0d0 - v))))
    else
        tmp = t_0 + ((-1.5d0) - (((r * w) * ((r * w) / (1.0d0 - v))) * 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 <= -1.5e+139) {
		tmp = t_0 + (-1.5 - (0.375 * ((r * (r * (w * w))) / (1.0 - v))));
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -1.5e+139:
		tmp = t_0 + (-1.5 - (0.375 * ((r * (r * (w * w))) / (1.0 - v))))
	else:
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 0.375))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -1.5e+139)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(1.0 - v)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))) * 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 <= -1.5e+139)
		tmp = t_0 + (-1.5 - (0.375 * ((r * (r * (w * w))) / (1.0 - v))));
	else
		tmp = t_0 + (-1.5 - (((r * w) * ((r * w) / (1.0 - v))) * 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[LessEqual[v, -1.5e+139], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.5e139

   1. Initial program 91.3%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in v around 0 86.6%

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

   if -1.5e139 < v

   1. Initial program 84.4%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in v around 0 74.5%

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

      1. add-sqr-sqrt 86.4%

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

      2. *-un-lft-identity 86.4%

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

      3. times-frac 86.4%

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

      4. associate-*r* 79.4%

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

      5. sqrt-prod 79.4%

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

      6. sqrt-prod 40.9%

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

      7. add-sqr-sqrt 72.9%

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

      8. sqrt-prod 38.2%

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

      9. add-sqr-sqrt 76.5%

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

      10. associate-*r* 65.7%

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

      11. sqrt-prod 65.7%

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

      12. sqrt-prod 37.2%

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

      13. add-sqr-sqrt 75.5%

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

      14. sqrt-prod 49.6%

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

      15. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 85.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right) \cdot 0.375\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 4 \cdot 10^{-156}:\\ \;\;\;\;\left(3 + t\_0\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.375 \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 4e-156)
     (- (+ 3.0 t_0) 4.5)
     (+ t_0 (- -1.5 (* 0.375 (/ (* r (* r (* w w))) (- 1.0 v))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 4e-156) {
		tmp = (3.0 + t_0) - 4.5;
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * (r * (w * w))) / (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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 4d-156) then
        tmp = (3.0d0 + t_0) - 4.5d0
    else
        tmp = t_0 + ((-1.5d0) - (0.375d0 * ((r * (r * (w * w))) / (1.0d0 - v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 4e-156) {
		tmp = (3.0 + t_0) - 4.5;
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * (r * (w * w))) / (1.0 - v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 4e-156:
		tmp = (3.0 + t_0) - 4.5
	else:
		tmp = t_0 + (-1.5 - (0.375 * ((r * (r * (w * w))) / (1.0 - v))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 4e-156)
		tmp = Float64(Float64(3.0 + t_0) - 4.5);
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(1.0 - v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 4e-156)
		tmp = (3.0 + t_0) - 4.5;
	else
		tmp = t_0 + (-1.5 - (0.375 * ((r * (r * (w * w))) / (1.0 - v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 4e-156], N[(N[(3.0 + t$95$0), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4.00000000000000016e-156

   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 79.3%

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

   5. Taylor expanded in r around 0 68.4%

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

   if 4.00000000000000016e-156 < r

   1. Initial program 86.2%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in v around 0 69.7%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-156}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 87.5%

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

5. Taylor expanded in v around 0 87.5%

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

   1. *-commutative 87.5%

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

7. Simplified 87.5%

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

   1. add-sqr-sqrt 87.4%

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

   2. *-un-lft-identity 87.4%

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

   3. times-frac 87.4%

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

   4. associate-*r* 80.9%

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

   5. sqrt-prod 80.9%

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

   6. sqrt-prod 41.9%

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

   7. add-sqr-sqrt 72.5%

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

   8. sqrt-prod 38.4%

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

   9. add-sqr-sqrt 75.6%

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

   10. associate-*r* 65.8%

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

   11. sqrt-prod 65.8%

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

   12. sqrt-prod 37.5%

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

   13. add-sqr-sqrt 75.6%

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

   14. sqrt-prod 50.2%

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

   15. add-sqr-sqrt 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 8: 56.8% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 80.3%

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

5. Taylor expanded in r around 0 61.7%

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

Alternative 9: 13.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

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

Java

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

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 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 80.3%

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

5. Taylor expanded in r around 0 61.7%

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

6. Taylor expanded in r around inf 13.8%

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

Reproduce

herbie shell --seed 2024139 
(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))