Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 99.1%

Time: 11.8s

Alternatives: 7

Speedup: 1.7×

• 54.1% 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: 84.9% 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: 99.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(w / N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision]), $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 97.6%

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

   1. div-inv 97.6%

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

   2. associate-*r* 99.8%

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

   3. associate-*l* 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in r around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/l* 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t_0\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(2 \cdot v - 3\right)\right)}{1 - v} \leq 3:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \frac{r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{1 - v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))) < 3

   1. Initial program 89.2%

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

   3. Simplified 92.7%

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

      1. associate-/r/ 92.8%

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

      2. associate-*r* 82.8%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-rgt-in 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*l* 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in r around 0 98.0%

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

   if 3 < (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)))

   1. Initial program 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in v around 0 79.5%

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

      1. *-commutative 79.5%

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

      2. unpow2 79.5%

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

      3. unpow2 79.5%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(2 \cdot v - 3\right)\right)}{1 - v} \leq 3:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \frac{r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{1 - v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := N[(-4.5 + N[(3.0 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 87.4%

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

   1. associate-*r* 97.5%

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

   2. *-commutative 97.5%

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

   3. *-un-lft-identity 97.5%

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

   4. associate-*r* 99.8%

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

   5. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := N[(N[(3.0 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision] * N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 87.4%

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

   1. associate-*r* 97.5%

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

   2. *-commutative 97.5%

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

   3. *-un-lft-identity 97.5%

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

   4. associate-*r* 99.8%

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

   5. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

   1. inv-pow 99.8%

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

   2. unpow-prod-down 99.8%

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

   3. inv-pow 99.8%

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

7. Applied egg-rr 99.8%

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

   1. unpow-1 99.8%

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

9. Simplified 99.8%

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

10. Taylor expanded in r around 0 99.8%

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

    1. associate-/r* 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 5: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.06 \cdot 10^{+64} \lor \neg \left(v \leq 0.4\right):\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1.06e+64) (not (<= v 0.4)))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* (* r w) 0.25)))))
     (+ -1.5 (+ t_0 (* (* (* r w) (* r w)) -0.375))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.06e+64) || !(v <= 0.4)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * ((r * w) * 0.25))));
	} else {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.375));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-1.06d+64)) .or. (.not. (v <= 0.4d0))) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * ((r * w) * 0.25d0))))
    else
        tmp = (-1.5d0) + (t_0 + (((r * w) * (r * w)) * (-0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.06e+64) || !(v <= 0.4)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * ((r * w) * 0.25))));
	} else {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1.06e+64) or not (v <= 0.4):
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * ((r * w) * 0.25))))
	else:
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.375))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1.06e+64) || !(v <= 0.4))
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(Float64(r * w) * 0.25)))));
	else
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -1.06e+64) || ~((v <= 0.4)))
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * ((r * w) * 0.25))));
	else
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1.06e+64], N[Not[LessEqual[v, 0.4]], $MachinePrecision]], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$0 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.06e64 or 0.40000000000000002 < v

   1. Initial program 83.6%

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

   3. Simplified 87.9%

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

      1. associate-/r/ 87.9%

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

      2. associate-*r* 83.5%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 99.8%

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

   if -1.06e64 < v < 0.40000000000000002

   1. Initial program 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in v around 0 78.9%

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

      1. *-commutative 78.9%

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

      2. unpow2 78.9%

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

      3. unpow2 78.9%

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

      4. swap-sqr 99.3%

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

      5. unpow2 99.3%

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

   6. Simplified 99.3%

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

      1. unpow2 99.3%

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

   8. Applied egg-rr 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.06 \cdot 10^{+64} \lor \neg \left(v \leq 0.4\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right)\\ \end{array} \]

Alternative 6: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.2 \cdot 10^{+64} \lor \neg \left(v \leq 0.02\right):\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1.2e+64) (not (<= v 0.02)))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* (* r w) 0.25)))))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* w (* r 0.375)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.2e+64) || !(v <= 0.02)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * ((r * w) * 0.25))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (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) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-1.2d+64)) .or. (.not. (v <= 0.02d0))) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * ((r * w) * 0.25d0))))
    else
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (w * (r * 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 <= -1.2e+64) || !(v <= 0.02)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * ((r * w) * 0.25))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1.2e+64) or not (v <= 0.02):
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * ((r * w) * 0.25))))
	else:
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1.2e+64) || !(v <= 0.02))
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(Float64(r * w) * 0.25)))));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 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 <= -1.2e+64) || ~((v <= 0.02)))
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * ((r * w) * 0.25))));
	else
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 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, -1.2e+64], N[Not[LessEqual[v, 0.02]], $MachinePrecision]], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.2e64 or 0.0200000000000000004 < v

   1. Initial program 83.6%

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

   3. Simplified 87.9%

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

      1. associate-/r/ 87.9%

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

      2. associate-*r* 83.5%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 99.8%

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

   if -1.2e64 < v < 0.0200000000000000004

   1. Initial program 86.9%

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

   3. Simplified 86.9%

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

      1. associate-/r/ 86.9%

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

      2. associate-*r* 79.4%

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

      3. swap-sqr 99.9%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in w around 0 99.3%

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

       1. *-commutative 99.3%

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

       2. associate-*r* 99.3%

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

       3. *-commutative 99.3%

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

       4. associate-*l* 99.4%

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

   11. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.2 \cdot 10^{+64} \lor \neg \left(v \leq 0.02\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 93.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $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 87.4%

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

4. Taylor expanded in v around 0 78.4%

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

   1. *-commutative 78.4%

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

   2. unpow2 78.4%

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

   3. unpow2 78.4%

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

   4. swap-sqr 94.6%

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

   5. unpow2 94.6%

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

6. Simplified 94.6%

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

   1. unpow2 94.6%

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

8. Applied egg-rr 94.6%

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

9. Final simplification 94.6%

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

Reproduce

herbie shell --seed 2023337 
(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))