Rosa's TurbineBenchmark

Percentage Accurate: 84.4% → 99.8%

Time: 10.1s

Alternatives: 7

Speedup: 1.7×

• 52.7% 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: 84.4% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.375 + v \cdot 0.25}{1 - v}\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+82}:\\ \;\;\;\;-1.5 + \left(t_1 + t_0 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_1 + t_0 \cdot \frac{w}{\frac{\frac{\frac{1}{w}}{r}}{r}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (+ -0.375 (* v 0.25)) (- 1.0 v))) (t_1 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e+82)
     (+ -1.5 (+ t_1 (* t_0 (* r (* w (* r w))))))
     (+ -1.5 (+ t_1 (* t_0 (/ w (/ (/ (/ 1.0 w) r) r))))))))

C

double code(double v, double w, double r) {
	double t_0 = (-0.375 + (v * 0.25)) / (1.0 - v);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+82) {
		tmp = -1.5 + (t_1 + (t_0 * (r * (w * (r * w)))));
	} else {
		tmp = -1.5 + (t_1 + (t_0 * (w / (((1.0 / w) / r) / r))));
	}
	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 = ((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)
    t_1 = 2.0d0 / (r * r)
    if ((w * w) <= 2d+82) then
        tmp = (-1.5d0) + (t_1 + (t_0 * (r * (w * (r * w)))))
    else
        tmp = (-1.5d0) + (t_1 + (t_0 * (w / (((1.0d0 / w) / r) / r))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (-0.375 + (v * 0.25)) / (1.0 - v);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+82) {
		tmp = -1.5 + (t_1 + (t_0 * (r * (w * (r * w)))));
	} else {
		tmp = -1.5 + (t_1 + (t_0 * (w / (((1.0 / w) / r) / r))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (-0.375 + (v * 0.25)) / (1.0 - v)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 2e+82:
		tmp = -1.5 + (t_1 + (t_0 * (r * (w * (r * w)))))
	else:
		tmp = -1.5 + (t_1 + (t_0 * (w / (((1.0 / w) / r) / r))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e+82)
		tmp = Float64(-1.5 + Float64(t_1 + Float64(t_0 * Float64(r * Float64(w * Float64(r * w))))));
	else
		tmp = Float64(-1.5 + Float64(t_1 + Float64(t_0 * Float64(w / Float64(Float64(Float64(1.0 / w) / r) / r)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (-0.375 + (v * 0.25)) / (1.0 - v);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 2e+82)
		tmp = -1.5 + (t_1 + (t_0 * (r * (w * (r * w)))));
	else
		tmp = -1.5 + (t_1 + (t_0 * (w / (((1.0 / w) / r) / r))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e+82], N[(-1.5 + N[(t$95$1 + N[(t$95$0 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$1 + N[(t$95$0 * N[(w / N[(N[(N[(1.0 / w), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.9999999999999999e82

   1. Initial program 92.6%

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

   3. Simplified 95.3%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. add-sqr-sqrt 73.5%

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

      4. pow2 73.5%

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

      5. *-commutative 73.5%

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

      6. associate-*r* 73.3%

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

      7. *-commutative 73.3%

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

      8. sqrt-prod 47.7%

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

      9. sqrt-prod 28.9%

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

      10. add-sqr-sqrt 50.9%

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

   5. Applied egg-rr 50.9%

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

      1. unpow2 50.9%

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

      2. *-commutative 50.9%

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

      3. associate-*r* 50.9%

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

      4. associate-*r* 50.9%

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

      5. add-sqr-sqrt 99.8%

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

      6. *-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 1.9999999999999999e82 < (*.f64 w w)

   1. Initial program 83.0%

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

   3. Simplified 86.2%

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

      1. associate-*r* 95.7%

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

      2. *-commutative 95.7%

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

      3. add-sqr-sqrt 49.5%

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

      4. pow2 49.5%

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

      5. *-commutative 49.5%

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

      6. associate-*r* 45.3%

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

      7. *-commutative 45.3%

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

      8. sqrt-prod 45.3%

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

      9. sqrt-prod 28.7%

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

      10. add-sqr-sqrt 49.5%

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

   5. Applied egg-rr 49.5%

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

      1. unpow2 49.5%

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

      2. *-commutative 49.5%

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

      3. associate-*r* 49.5%

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

      4. associate-*r* 49.5%

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

      5. add-sqr-sqrt 95.7%

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

      6. *-commutative 95.7%

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

   7. Applied egg-rr 95.7%

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

      1. *-commutative 95.7%

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

      2. /-rgt-identity 95.7%

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

      3. clear-num 95.7%

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

      4. associate-/r/ 95.7%

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

      5. associate-/r* 95.7%

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

      6. un-div-inv 95.7%

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

      7. div-inv 95.6%

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

      8. un-div-inv 95.6%

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

      9. associate-/l* 99.9%

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

      10. *-commutative 99.9%

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

      11. associate-/l* 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+82}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \frac{w}{\frac{\frac{\frac{1}{w}}{r}}{r}}\right)\\ \end{array} \]

Alternative 2: 91.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 2 \cdot 10^{+128}:\\ \;\;\;\;-1.5 + \left(t_0 + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + -1.5\right) - 0.375 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 2e+128)
     (+
      -1.5
      (+ t_0 (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r (* r (* w w))))))
     (- (+ t_0 -1.5) (* 0.375 (* w (* r (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 2e+128) {
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = (t_0 + -1.5) - (0.375 * (w * (r * (r * w))));
	}
	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 (w <= 2d+128) then
        tmp = (-1.5d0) + (t_0 + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r * (r * (w * w)))))
    else
        tmp = (t_0 + (-1.5d0)) - (0.375d0 * (w * (r * (r * w))))
    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 (w <= 2e+128) {
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = (t_0 + -1.5) - (0.375 * (w * (r * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if w <= 2e+128:
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	else:
		tmp = (t_0 + -1.5) - (0.375 * (w * (r * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (w <= 2e+128)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))));
	else
		tmp = Float64(Float64(t_0 + -1.5) - Float64(0.375 * Float64(w * Float64(r * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (w <= 2e+128)
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	else
		tmp = (t_0 + -1.5) - (0.375 * (w * (r * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 2e+128], N[(-1.5 + N[(t$95$0 + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + -1.5), $MachinePrecision] - N[(0.375 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 2.0000000000000002e128

   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 92.5%

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

   if 2.0000000000000002e128 < w

   1. Initial program 82.9%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in v around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. unpow2 82.9%

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

      3. unpow2 82.9%

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

      4. swap-sqr 100.0%

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

      5. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 2 \cdot 10^{+128}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - 0.375 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 3: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 2 \cdot 10^{+163}:\\ \;\;\;\;-1.5 + \left(t_0 + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + -1.5\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.375\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 2e+163)
     (+
      -1.5
      (+ t_0 (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r (* w (* r w))))))
     (- (+ t_0 -1.5) (* (* (* 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 (w <= 2e+163) {
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	} else {
		tmp = (t_0 + -1.5) - (((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 (w <= 2d+163) then
        tmp = (-1.5d0) + (t_0 + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r * (w * (r * w)))))
    else
        tmp = (t_0 + (-1.5d0)) - (((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 (w <= 2e+163) {
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	} else {
		tmp = (t_0 + -1.5) - (((r * w) * (r * w)) * 0.375);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if w <= 2e+163:
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (r * w)))))
	else:
		tmp = (t_0 + -1.5) - (((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 (w <= 2e+163)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r * Float64(w * Float64(r * w))))));
	else
		tmp = Float64(Float64(t_0 + -1.5) - 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 (w <= 2e+163)
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	else
		tmp = (t_0 + -1.5) - (((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[w, 2e+163], N[(-1.5 + N[(t$95$0 + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + -1.5), $MachinePrecision] - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 1.9999999999999999e163

   1. Initial program 88.5%

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

   3. Simplified 91.8%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

      3. add-sqr-sqrt 63.7%

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

      4. pow2 63.7%

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

      5. *-commutative 63.7%

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

      6. associate-*r* 61.4%

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

      7. *-commutative 61.4%

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

      8. sqrt-prod 45.5%

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

      9. sqrt-prod 25.4%

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

      10. add-sqr-sqrt 49.6%

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

   5. Applied egg-rr 49.6%

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

      1. unpow2 49.6%

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

      2. *-commutative 49.6%

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

      3. associate-*r* 49.6%

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

      4. associate-*r* 49.6%

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

      5. add-sqr-sqrt 98.6%

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

      6. *-commutative 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 1.9999999999999999e163 < w

   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 93.1%

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

   4. Taylor expanded in v around 0 86.2%

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

      1. *-commutative 86.2%

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

      2. unpow2 86.2%

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

      3. unpow2 86.2%

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

      4. swap-sqr 100.0%

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

      5. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. unpow2 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 2 \cdot 10^{+163}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.375\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * ((r * w) / ((1.0 / w) / r)))) + -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)) / (1.0d0 - v)) * ((r * w) / ((1.0d0 / w) / r)))) + (-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)) / (1.0 - v)) * ((r * w) / ((1.0 / w) / r)))) + -1.5;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 91.2%

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

   1. associate-*r* 97.9%

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

   2. *-commutative 97.9%

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

   3. add-sqr-sqrt 62.7%

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

   4. pow2 62.7%

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

   5. *-commutative 62.7%

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

   6. associate-*r* 60.7%

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

   7. *-commutative 60.7%

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

   8. sqrt-prod 46.6%

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

   9. sqrt-prod 28.8%

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

   10. add-sqr-sqrt 50.3%

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

5. Applied egg-rr 50.3%

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

   1. unpow2 50.3%

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

   2. *-commutative 50.3%

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

   3. associate-*r* 50.3%

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

   4. associate-*r* 50.3%

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

   5. add-sqr-sqrt 97.9%

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

   6. *-commutative 97.9%

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

7. Applied egg-rr 97.9%

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

   1. *-commutative 97.9%

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

   2. /-rgt-identity 97.9%

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

   3. clear-num 97.9%

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

   4. associate-/r/ 97.9%

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

   5. associate-/r* 97.9%

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

   6. un-div-inv 97.9%

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

   7. div-inv 97.9%

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

   8. un-div-inv 97.9%

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

   9. associate-/l* 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 91.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 97.9%

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

4. Taylor expanded in v around 0 83.8%

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

   1. *-commutative 83.8%

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

   2. unpow2 83.8%

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

   3. unpow2 83.8%

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

   4. swap-sqr 93.3%

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

   5. unpow2 93.3%

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

6. Simplified 93.3%

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

   1. unpow2 93.3%

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

8. Applied egg-rr 93.3%

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

   1. associate-*l* 93.3%

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

   2. *-commutative 93.3%

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

   3. remove-double-div 93.3%

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

   4. associate-/l/ 93.3%

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

   5. div-inv 93.3%

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

   6. associate-/l* 93.3%

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

10. Applied egg-rr 93.3%

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

    1. div-inv 93.3%

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

    2. *-commutative 93.3%

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

    3. clear-num 93.3%

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

    4. associate-*l* 92.7%

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

    5. clear-num 92.7%

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

    6. associate-/l/ 92.7%

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

    7. associate-/r/ 92.7%

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

    8. clear-num 92.7%

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

    9. /-rgt-identity 92.7%

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

    10. *-commutative 92.7%

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

12. Applied egg-rr 92.7%

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

13. Final simplification 92.7%

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

Alternative 6: 92.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 97.9%

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

4. Taylor expanded in v around 0 83.8%

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

   1. *-commutative 83.8%

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

   2. unpow2 83.8%

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

   3. unpow2 83.8%

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

   4. swap-sqr 93.3%

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

   5. unpow2 93.3%

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

6. Simplified 93.3%

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

   1. unpow2 93.3%

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

8. Applied egg-rr 93.3%

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

   1. associate-*l* 93.3%

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

   2. *-commutative 93.3%

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

   3. remove-double-div 93.3%

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

   4. associate-/l/ 93.3%

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

   5. div-inv 93.3%

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

   6. associate-/l* 93.3%

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

10. Applied egg-rr 93.3%

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

    1. clear-num 93.3%

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

    2. associate-/r/ 93.3%

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

    3. clear-num 93.3%

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

    4. associate-/l/ 93.3%

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

    5. associate-/r/ 93.3%

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

    6. metadata-eval 93.3%

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

    7. associate-*r* 93.3%

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

    8. associate-*l* 92.7%

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

    9. *-commutative 92.7%

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

    10. associate-*l* 92.7%

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

    11. associate-*l* 93.3%

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

12. Applied egg-rr 93.3%

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

13. Final simplification 93.3%

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

Alternative 7: 92.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot w}{\frac{\frac{2.6666666666666665}{w}}{r}} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (- (+ (/ 2.0 (* r r)) -1.5) (/ (* r w) (/ (/ 2.6666666666666665 w) r))))

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + -1.5) - ((r * w) / ((2.6666666666666665 / w) / r));
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / (r * r)) + (-1.5d0)) - ((r * w) / ((2.6666666666666665d0 / w) / r))
end function

Java

public static double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + -1.5) - ((r * w) / ((2.6666666666666665 / w) / r));
}

Python

def code(v, w, r):
	return ((2.0 / (r * r)) + -1.5) - ((r * w) / ((2.6666666666666665 / w) / r))

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / Float64(r * r)) + -1.5) - Float64(Float64(r * w) / Float64(Float64(2.6666666666666665 / w) / r)))
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((2.0 / (r * r)) + -1.5) - ((r * w) / ((2.6666666666666665 / w) / r));
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] / N[(N[(2.6666666666666665 / w), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 97.9%

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

4. Taylor expanded in v around 0 83.8%

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

   1. *-commutative 83.8%

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

   2. unpow2 83.8%

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

   3. unpow2 83.8%

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

   4. swap-sqr 93.3%

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

   5. unpow2 93.3%

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

6. Simplified 93.3%

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

   1. unpow2 93.3%

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

8. Applied egg-rr 93.3%

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

   1. associate-*l* 93.3%

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

   2. *-commutative 93.3%

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

   3. remove-double-div 93.3%

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

   4. associate-/l/ 93.3%

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

   5. div-inv 93.3%

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

   6. associate-/l* 93.3%

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

10. Applied egg-rr 93.3%

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

11. Taylor expanded in w around 0 93.3%

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot w}{\color{blue}{\frac{2.6666666666666665}{r \cdot w}}} \]
12. Step-by-step derivation

    1. *-commutative 93.3%

       \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot w}{\frac{2.6666666666666665}{\color{blue}{w \cdot r}}} \]

    2. associate-/r* 93.3%

       \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot w}{\color{blue}{\frac{\frac{2.6666666666666665}{w}}{r}}} \]

13. Simplified 93.3%

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot w}{\color{blue}{\frac{\frac{2.6666666666666665}{w}}{r}}} \]

14. Final simplification 93.3%

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot w}{\frac{\frac{2.6666666666666665}{w}}{r}} \]

Reproduce

herbie shell --seed 2023321 
(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))