Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 98.9%

Time: 14.5s

Alternatives: 7

Speedup: 1.7×

• 54.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;-1.5 + \left(2 \cdot {r}^{-2} + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + r \cdot \left(w \cdot \frac{r \cdot w}{\frac{1 - v}{\mathsf{fma}\left(v, 0.25, -0.375\right)}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (if (<= r 2.5e-34)
   (+ -1.5 (+ (* 2.0 (pow r -2.0)) (* -0.25 (* w (* r (* r w))))))
   (+
    (/ 2.0 (* r r))
    (+ -1.5 (* r (* w (/ (* r w) (/ (- 1.0 v) (fma v 0.25 -0.375)))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.5e-34) {
		tmp = -1.5 + ((2.0 * pow(r, -2.0)) + (-0.25 * (w * (r * (r * w)))));
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 + (r * (w * ((r * w) / ((1.0 - v) / fma(v, 0.25, -0.375))))));
	}
	return tmp;
}

Julia

r = abs(r)
function code(v, w, r)
	tmp = 0.0
	if (r <= 2.5e-34)
		tmp = Float64(-1.5 + Float64(Float64(2.0 * (r ^ -2.0)) + Float64(-0.25 * Float64(w * Float64(r * Float64(r * w))))));
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(r * Float64(w * Float64(Float64(r * w) / Float64(Float64(1.0 - v) / fma(v, 0.25, -0.375)))))));
	end
	return tmp
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := If[LessEqual[r, 2.5e-34], N[(-1.5 + N[(N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(r * N[(w * N[(N[(r * w), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(v * 0.25 + -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.5000000000000001e-34

   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 84.4%

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

   4. Taylor expanded in v around inf 78.5%

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

      1. unpow2 78.5%

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

      2. unpow2 78.5%

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

      3. swap-sqr 95.6%

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

      4. unpow2 95.6%

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

   6. Simplified 95.6%

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

      1. unpow2 95.6%

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

      2. associate-*r* 94.4%

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

   8. Applied egg-rr 94.4%

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

      1. div-inv 94.4%

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

      2. *-commutative 94.4%

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

      3. pow2 94.4%

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

      4. pow-flip 94.5%

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

      5. metadata-eval 94.5%

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

   10. Applied egg-rr 94.5%

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

   if 2.5000000000000001e-34 < r

   1. Initial program 91.5%

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

   3. Simplified 89.7%

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

      1. associate-/r* 89.7%

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

      2. *-un-lft-identity 89.7%

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

      3. associate-*r* 91.0%

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

      4. times-frac 90.8%

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

   5. Applied egg-rr 90.8%

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

      1. frac-times 91.0%

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

      2. *-un-lft-identity 91.0%

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

      3. associate-/r/ 91.1%

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

      4. *-commutative 91.1%

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

      5. associate-*l/ 99.7%

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

      6. fma-def 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*l* 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. associate-*r* 99.7%

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;-1.5 + \left(2 \cdot {r}^{-2} + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + r \cdot \left(w \cdot \frac{r \cdot w}{\frac{1 - v}{\mathsf{fma}\left(v, 0.25, -0.375\right)}}\right)\right)\\ \end{array} \]

Alternative 2: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-34}:\\ \;\;\;\;-1.5 + \left(2 \cdot {r}^{-2} + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\frac{r}{1 - v} \cdot \left(\mathsf{fma}\left(v, 0.25, -0.375\right) \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (if (<= r 2e-34)
   (+ -1.5 (+ (* 2.0 (pow r -2.0)) (* -0.25 (* w (* r (* r w))))))
   (+
    (/ 2.0 (* r r))
    (+ -1.5 (* (* r w) (* (/ r (- 1.0 v)) (* (fma v 0.25 -0.375) w)))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-34) {
		tmp = -1.5 + ((2.0 * pow(r, -2.0)) + (-0.25 * (w * (r * (r * w)))));
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 + ((r * w) * ((r / (1.0 - v)) * (fma(v, 0.25, -0.375) * w))));
	}
	return tmp;
}

Julia

r = abs(r)
function code(v, w, r)
	tmp = 0.0
	if (r <= 2e-34)
		tmp = Float64(-1.5 + Float64(Float64(2.0 * (r ^ -2.0)) + Float64(-0.25 * Float64(w * Float64(r * Float64(r * w))))));
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(r * w) * Float64(Float64(r / Float64(1.0 - v)) * Float64(fma(v, 0.25, -0.375) * w)))));
	end
	return tmp
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := If[LessEqual[r, 2e-34], N[(-1.5 + N[(N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(N[(v * 0.25 + -0.375), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.99999999999999986e-34

   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 84.4%

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

   4. Taylor expanded in v around inf 78.5%

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

      1. unpow2 78.5%

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

      2. unpow2 78.5%

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

      3. swap-sqr 95.6%

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

      4. unpow2 95.6%

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

   6. Simplified 95.6%

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

      1. unpow2 95.6%

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

      2. associate-*r* 94.4%

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

   8. Applied egg-rr 94.4%

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

      1. div-inv 94.4%

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

      2. *-commutative 94.4%

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

      3. pow2 94.4%

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

      4. pow-flip 94.5%

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

      5. metadata-eval 94.5%

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

   10. Applied egg-rr 94.5%

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

   if 1.99999999999999986e-34 < r

   1. Initial program 91.5%

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

   3. Simplified 89.7%

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

      1. associate-/r* 89.7%

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

      2. associate-/r/ 89.7%

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

      3. *-commutative 89.7%

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

      4. associate-*l/ 97.5%

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

      5. associate-*r* 97.4%

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

      6. add-sqr-sqrt 97.4%

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

      7. associate-*r* 97.4%

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

      8. fma-def 97.4%

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

      9. associate-*r* 79.5%

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

      10. sqrt-prod 79.6%

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

      11. sqrt-prod 97.4%

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

      12. add-sqr-sqrt 97.5%

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

      13. sqrt-prod 57.2%

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

      14. add-sqr-sqrt 66.3%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in r around 0 98.3%

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

      1. associate-/l* 99.8%

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

      2. associate-/r/ 99.7%

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

      3. sub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. metadata-eval 99.7%

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

      6. fma-udef 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-34}:\\ \;\;\;\;-1.5 + \left(2 \cdot {r}^{-2} + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\frac{r}{1 - v} \cdot \left(\mathsf{fma}\left(v, 0.25, -0.375\right) \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.2× speedup

Math

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

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (+ (* (* r w) (* (/ (fma v 0.25 -0.375) (- 1.0 v)) (* r w))) -1.5)))

C

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

Julia

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

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(N[(N[(v * 0.25 + -0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

3. Simplified 81.8%

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

   1. associate-/r* 83.0%

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

   2. associate-/r/ 83.0%

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

   3. *-commutative 83.0%

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

   4. associate-*l/ 87.8%

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

   5. associate-*r* 87.8%

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

   6. add-sqr-sqrt 87.7%

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

   7. associate-*r* 87.7%

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

   8. fma-def 87.7%

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

   9. associate-*r* 80.3%

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

   10. sqrt-prod 80.3%

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

   11. sqrt-prod 40.4%

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

   12. add-sqr-sqrt 69.2%

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

   13. sqrt-prod 33.2%

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

   14. add-sqr-sqrt 68.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2100000000 \lor \neg \left(v \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \frac{w \cdot \left(r \cdot 0.375\right)}{\frac{\frac{-1}{r}}{w}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -2100000000.0) (not (<= v 2.6e-9)))
     (+ t_0 (+ -1.5 (* (* r w) (* w (* r -0.25)))))
     (+ t_0 (+ -1.5 (/ (* w (* r 0.375)) (/ (/ -1.0 r) w)))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2100000000.0) || !(v <= 2.6e-9)) {
		tmp = t_0 + (-1.5 + ((r * w) * (w * (r * -0.25))));
	} else {
		tmp = t_0 + (-1.5 + ((w * (r * 0.375)) / ((-1.0 / r) / w)));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
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 <= (-2100000000.0d0)) .or. (.not. (v <= 2.6d-9))) then
        tmp = t_0 + ((-1.5d0) + ((r * w) * (w * (r * (-0.25d0)))))
    else
        tmp = t_0 + ((-1.5d0) + ((w * (r * 0.375d0)) / (((-1.0d0) / r) / w)))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2100000000.0) || !(v <= 2.6e-9)) {
		tmp = t_0 + (-1.5 + ((r * w) * (w * (r * -0.25))));
	} else {
		tmp = t_0 + (-1.5 + ((w * (r * 0.375)) / ((-1.0 / r) / w)));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2100000000.0) or not (v <= 2.6e-9):
		tmp = t_0 + (-1.5 + ((r * w) * (w * (r * -0.25))))
	else:
		tmp = t_0 + (-1.5 + ((w * (r * 0.375)) / ((-1.0 / r) / w)))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2100000000.0) || !(v <= 2.6e-9))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(w * Float64(r * -0.25)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(w * Float64(r * 0.375)) / Float64(Float64(-1.0 / r) / w))));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2100000000.0) || ~((v <= 2.6e-9)))
		tmp = t_0 + (-1.5 + ((r * w) * (w * (r * -0.25))));
	else
		tmp = t_0 + (-1.5 + ((w * (r * 0.375)) / ((-1.0 / r) / w)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -2100000000.0], N[Not[LessEqual[v, 2.6e-9]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.1e9 or 2.6000000000000001e-9 < v

   1. Initial program 80.0%

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

   3. Simplified 73.6%

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

      1. associate-/r* 75.9%

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

      2. associate-/r/ 76.0%

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

      3. *-commutative 76.0%

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

      4. associate-*l/ 85.0%

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

      5. associate-*r* 85.0%

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

      6. add-sqr-sqrt 84.9%

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

      7. associate-*r* 84.9%

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

      8. fma-def 84.9%

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

      9. associate-*r* 76.6%

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

      10. sqrt-prod 76.7%

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

      11. sqrt-prod 41.1%

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

      12. add-sqr-sqrt 70.2%

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

      13. sqrt-prod 35.3%

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

      14. add-sqr-sqrt 67.2%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 99.8%

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

      1. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   if -2.1e9 < v < 2.6000000000000001e-9

   1. Initial program 90.9%

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

   3. Simplified 90.9%

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

      1. associate-/r* 90.9%

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

      2. associate-/r/ 90.9%

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

      3. *-commutative 90.9%

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

      4. associate-*l/ 90.9%

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

      5. associate-*r* 90.9%

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

      6. add-sqr-sqrt 90.8%

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

      7. associate-*r* 90.9%

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

      8. fma-def 90.9%

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

      9. associate-*r* 84.4%

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

      10. sqrt-prod 84.4%

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

      11. sqrt-prod 39.6%

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

      12. add-sqr-sqrt 68.0%

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

      13. sqrt-prod 30.7%

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

      14. add-sqr-sqrt 70.6%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around 0 99.8%

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

      1. associate-*r* 99.7%

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

   8. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 98.1%

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

      3. /-rgt-identity 98.1%

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

      4. *-commutative 98.1%

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

      5. associate-*l* 95.9%

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

      6. associate-/r/ 95.9%

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

      7. associate-/r* 95.9%

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

      8. frac-2neg 95.9%

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

      9. associate-*l/ 99.8%

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

      10. distribute-rgt-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. metadata-eval 99.8%

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

   10. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2100000000 \lor \neg \left(v \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{w \cdot \left(r \cdot 0.375\right)}{\frac{\frac{-1}{r}}{w}}\right)\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2.6 \cdot 10^{+17} \lor \neg \left(v \leq 10^{+62}\right):\\ \;\;\;\;-1.5 + \left(t_0 + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(-0.375 \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -2.6e+17) (not (<= v 1e+62)))
     (+ -1.5 (+ t_0 (* -0.25 (* w (* r (* r w))))))
     (+ t_0 (+ -1.5 (* (* r w) (* -0.375 (* r w))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2.6e+17) || !(v <= 1e+62)) {
		tmp = -1.5 + (t_0 + (-0.25 * (w * (r * (r * w)))));
	} else {
		tmp = t_0 + (-1.5 + ((r * w) * (-0.375 * (r * w))));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
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 <= (-2.6d+17)) .or. (.not. (v <= 1d+62))) then
        tmp = (-1.5d0) + (t_0 + ((-0.25d0) * (w * (r * (r * w)))))
    else
        tmp = t_0 + ((-1.5d0) + ((r * w) * ((-0.375d0) * (r * w))))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2.6e+17) || !(v <= 1e+62)) {
		tmp = -1.5 + (t_0 + (-0.25 * (w * (r * (r * w)))));
	} else {
		tmp = t_0 + (-1.5 + ((r * w) * (-0.375 * (r * w))));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2.6e+17) or not (v <= 1e+62):
		tmp = -1.5 + (t_0 + (-0.25 * (w * (r * (r * w)))))
	else:
		tmp = t_0 + (-1.5 + ((r * w) * (-0.375 * (r * w))))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2.6e+17) || !(v <= 1e+62))
		tmp = Float64(-1.5 + Float64(t_0 + Float64(-0.25 * Float64(w * Float64(r * Float64(r * w))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(-0.375 * Float64(r * w)))));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2.6e+17) || ~((v <= 1e+62)))
		tmp = -1.5 + (t_0 + (-0.25 * (w * (r * (r * w)))));
	else
		tmp = t_0 + (-1.5 + ((r * w) * (-0.375 * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -2.6e+17], N[Not[LessEqual[v, 1e+62]], $MachinePrecision]], N[(-1.5 + N[(t$95$0 + N[(-0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(-0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.6e17 or 1.00000000000000004e62 < v

   1. Initial program 80.1%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in v around inf 77.1%

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

      1. unpow2 77.1%

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

      2. unpow2 77.1%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. associate-*r* 97.4%

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

   8. Applied egg-rr 97.4%

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

   if -2.6e17 < v < 1.00000000000000004e62

   1. Initial program 89.4%

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

   3. Simplified 89.4%

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

      1. associate-/r* 89.3%

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

      2. associate-/r/ 89.4%

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

      3. *-commutative 89.4%

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

      4. associate-*l/ 89.6%

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

      5. associate-*r* 89.6%

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

      6. add-sqr-sqrt 89.5%

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

      7. associate-*r* 89.5%

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

      8. fma-def 89.5%

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

      9. associate-*r* 83.0%

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

      10. sqrt-prod 83.0%

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

      11. sqrt-prod 41.6%

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

      12. add-sqr-sqrt 69.8%

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

      13. sqrt-prod 31.9%

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

      14. add-sqr-sqrt 70.6%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around 0 98.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.6 \cdot 10^{+17} \lor \neg \left(v \leq 10^{+62}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(-0.375 \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 6: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2100000000 \lor \neg \left(v \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(-0.375 \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -2100000000.0) (not (<= v 2e-9)))
     (+ t_0 (+ -1.5 (* (* r w) (* w (* r -0.25)))))
     (+ t_0 (+ -1.5 (* (* r w) (* -0.375 (* r w))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2100000000.0) || !(v <= 2e-9)) {
		tmp = t_0 + (-1.5 + ((r * w) * (w * (r * -0.25))));
	} else {
		tmp = t_0 + (-1.5 + ((r * w) * (-0.375 * (r * w))));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
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 <= (-2100000000.0d0)) .or. (.not. (v <= 2d-9))) then
        tmp = t_0 + ((-1.5d0) + ((r * w) * (w * (r * (-0.25d0)))))
    else
        tmp = t_0 + ((-1.5d0) + ((r * w) * ((-0.375d0) * (r * w))))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2100000000.0) || !(v <= 2e-9)) {
		tmp = t_0 + (-1.5 + ((r * w) * (w * (r * -0.25))));
	} else {
		tmp = t_0 + (-1.5 + ((r * w) * (-0.375 * (r * w))));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2100000000.0) or not (v <= 2e-9):
		tmp = t_0 + (-1.5 + ((r * w) * (w * (r * -0.25))))
	else:
		tmp = t_0 + (-1.5 + ((r * w) * (-0.375 * (r * w))))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2100000000.0) || !(v <= 2e-9))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(w * Float64(r * -0.25)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(-0.375 * Float64(r * w)))));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2100000000.0) || ~((v <= 2e-9)))
		tmp = t_0 + (-1.5 + ((r * w) * (w * (r * -0.25))));
	else
		tmp = t_0 + (-1.5 + ((r * w) * (-0.375 * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -2100000000.0], N[Not[LessEqual[v, 2e-9]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(-0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.1e9 or 2.00000000000000012e-9 < v

   1. Initial program 80.0%

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

   3. Simplified 73.6%

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

      1. associate-/r* 75.9%

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

      2. associate-/r/ 76.0%

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

      3. *-commutative 76.0%

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

      4. associate-*l/ 85.0%

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

      5. associate-*r* 85.0%

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

      6. add-sqr-sqrt 84.9%

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

      7. associate-*r* 84.9%

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

      8. fma-def 84.9%

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

      9. associate-*r* 76.6%

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

      10. sqrt-prod 76.7%

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

      11. sqrt-prod 41.1%

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

      12. add-sqr-sqrt 70.2%

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

      13. sqrt-prod 35.3%

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

      14. add-sqr-sqrt 67.2%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 99.8%

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

      1. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   if -2.1e9 < v < 2.00000000000000012e-9

   1. Initial program 90.9%

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

   3. Simplified 90.9%

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

      1. associate-/r* 90.9%

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

      2. associate-/r/ 90.9%

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

      3. *-commutative 90.9%

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

      4. associate-*l/ 90.9%

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

      5. associate-*r* 90.9%

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

      6. add-sqr-sqrt 90.8%

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

      7. associate-*r* 90.9%

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

      8. fma-def 90.9%

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

      9. associate-*r* 84.4%

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

      10. sqrt-prod 84.4%

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

      11. sqrt-prod 39.6%

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

      12. add-sqr-sqrt 68.0%

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

      13. sqrt-prod 30.7%

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

      14. add-sqr-sqrt 70.6%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2100000000 \lor \neg \left(v \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(-0.375 \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 7: 91.7% accurate, 1.7× speedup

Math

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

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (+ -1.5 (+ (/ 2.0 (* r r)) (* -0.25 (* w (* r (* r w)))))))

C

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

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (-1.5d0) + ((2.0d0 / (r * r)) + ((-0.25d0) * (w * (r * (r * w)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

3. Simplified 87.8%

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

4. Taylor expanded in v around inf 76.6%

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

   1. unpow2 76.6%

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

   2. unpow2 76.6%

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

   3. swap-sqr 93.3%

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

   4. unpow2 93.3%

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

6. Simplified 93.3%

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

   1. unpow2 93.3%

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

   2. associate-*r* 91.2%

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

8. Applied egg-rr 91.2%

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

9. Final simplification 91.2%

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

Reproduce

herbie shell --seed 2023305 
(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))