Rosa's TurbineBenchmark

Percentage Accurate: 84.0% → 99.8%

Time: 16.9s

Alternatives: 11

Speedup: 1.7×

• 53.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{+230}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - \left(0.125 \cdot \left(r \cdot w\right)\right) \cdot \frac{3 + -2 \cdot v}{\frac{\frac{1 - v}{w}}{r}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 5e+230)
     (+
      t_0
      (- -1.5 (* (* r (* w (* r w))) (/ (+ 0.375 (* v -0.25)) (- 1.0 v)))))
     (+
      -4.5
      (-
       (+ 3.0 t_0)
       (* (* 0.125 (* r w)) (/ (+ 3.0 (* -2.0 v)) (/ (/ (- 1.0 v) w) r))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 5e+230) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - ((0.125 * (r * w)) * ((3.0 + (-2.0 * v)) / (((1.0 - v) / w) / 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) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((w * w) <= 5d+230) then
        tmp = t_0 + ((-1.5d0) - ((r * (w * (r * w))) * ((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v))))
    else
        tmp = (-4.5d0) + ((3.0d0 + t_0) - ((0.125d0 * (r * w)) * ((3.0d0 + ((-2.0d0) * v)) / (((1.0d0 - v) / w) / r))))
    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 * w) <= 5e+230) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - ((0.125 * (r * w)) * ((3.0 + (-2.0 * v)) / (((1.0 - v) / w) / r))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 5e+230:
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))))
	else:
		tmp = -4.5 + ((3.0 + t_0) - ((0.125 * (r * w)) * ((3.0 + (-2.0 * v)) / (((1.0 - v) / w) / r))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 5e+230)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)))));
	else
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(Float64(0.125 * Float64(r * w)) * Float64(Float64(3.0 + Float64(-2.0 * v)) / Float64(Float64(Float64(1.0 - v) / w) / r)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 5e+230)
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	else
		tmp = -4.5 + ((3.0 + t_0) - ((0.125 * (r * w)) * ((3.0 + (-2.0 * v)) / (((1.0 - v) / w) / r))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 5e+230], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(0.125 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 - v), $MachinePrecision] / w), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 5.0000000000000003e230

   1. Initial program 89.6%

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

      1. associate--l- 89.6%

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

      2. +-commutative 89.6%

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

      3. associate--l+ 89.6%

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

      4. +-commutative 89.6%

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

      5. associate--r+ 89.6%

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

      6. metadata-eval 89.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 91.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 91.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 91.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 91.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 92.9%

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

   4. Taylor expanded in r around 0 92.9%

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

      1. unpow2 92.9%

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

      2. associate-*l* 99.3%

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

   6. Simplified 99.3%

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

   if 5.0000000000000003e230 < (*.f64 w w)

   1. Initial program 72.2%

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

   3. Simplified 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. add-sqr-sqrt 73.3%

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

      3. times-frac 73.3%

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

      4. unswap-sqr 73.3%

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

      5. sqrt-prod 37.9%

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

      6. add-sqr-sqrt 41.6%

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

      7. unswap-sqr 67.0%

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

      8. sqrt-prod 50.5%

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

      9. add-sqr-sqrt 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. times-frac 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate-/r/ 99.9%

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

      2. metadata-eval 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r* 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{+230}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.125 \cdot \left(r \cdot w\right)\right) \cdot \frac{3 + -2 \cdot v}{\frac{\frac{1 - v}{w}}{r}}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Simplified 76.9%

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

   1. *-un-lft-identity 76.9%

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

   2. add-sqr-sqrt 76.9%

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

   3. times-frac 76.9%

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

   4. unswap-sqr 76.9%

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

   5. sqrt-prod 41.6%

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

   6. add-sqr-sqrt 58.7%

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

   7. unswap-sqr 77.1%

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

   8. sqrt-prod 53.4%

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

   9. add-sqr-sqrt 99.0%

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

5. Applied egg-rr 99.0%

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

6. Final simplification 99.0%

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

Alternative 3: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 10^{+183}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{4}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 1e+183)
     (+
      t_0
      (- -1.5 (* (* r (* w (* r w))) (/ (+ 0.375 (* v -0.25)) (- 1.0 v)))))
     (+ -4.5 (- (+ 3.0 t_0) (* w (* (* r w) (/ r 4.0))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 1e+183) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (w * ((r * w) * (r / 4.0))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (w <= 1d+183) then
        tmp = t_0 + ((-1.5d0) - ((r * (w * (r * w))) * ((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v))))
    else
        tmp = (-4.5d0) + ((3.0d0 + t_0) - (w * ((r * w) * (r / 4.0d0))))
    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 <= 1e+183) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (w * ((r * w) * (r / 4.0))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if w <= 1e+183:
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))))
	else:
		tmp = -4.5 + ((3.0 + t_0) - (w * ((r * w) * (r / 4.0))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (w <= 1e+183)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)))));
	else
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(w * Float64(Float64(r * w) * Float64(r / 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (w <= 1e+183)
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	else
		tmp = -4.5 + ((3.0 + t_0) - (w * ((r * w) * (r / 4.0))));
	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, 1e+183], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(w * N[(N[(r * w), $MachinePrecision] * N[(r / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 9.99999999999999947e182

   1. Initial program 85.1%

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

      1. associate--l- 85.1%

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

      2. +-commutative 85.1%

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

      3. associate--l+ 85.1%

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

      4. +-commutative 85.1%

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

      5. associate--r+ 85.1%

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

      6. metadata-eval 85.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 87.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 87.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 87.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 87.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 88.0%

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

   4. Taylor expanded in r around 0 88.0%

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

      1. unpow2 88.0%

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

      2. associate-*l* 98.1%

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

   6. Simplified 98.1%

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

   if 9.99999999999999947e182 < w

   1. Initial program 74.8%

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

   3. Simplified 74.8%

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

      1. *-un-lft-identity 74.8%

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

      2. add-sqr-sqrt 74.8%

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

      3. times-frac 74.8%

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

      4. unswap-sqr 74.8%

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

      5. sqrt-prod 42.2%

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

      6. add-sqr-sqrt 42.5%

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

      7. unswap-sqr 64.4%

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

      8. sqrt-prod 51.6%

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

      9. add-sqr-sqrt 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. times-frac 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate-/r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r* 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in v around inf 74.8%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)}\right) + -4.5 \]
   11. Step-by-step derivation

       1. unpow2 74.8%

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

       2. associate-*r* 74.8%

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

       3. unpow2 74.8%

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

       4. associate-*r* 74.8%

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

       5. associate-*r* 74.8%

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

       6. metadata-eval 74.8%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\left(\color{blue}{\frac{1}{4}} \cdot w\right) \cdot w\right) \cdot r\right) \cdot r\right) + -4.5 \]

       7. associate-/r/ 74.8%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\color{blue}{\frac{1}{\frac{4}{w}}} \cdot w\right) \cdot r\right) \cdot r\right) + -4.5 \]

       8. associate-*r* 87.6%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{1}{\frac{4}{w}} \cdot \left(w \cdot r\right)\right)} \cdot r\right) + -4.5 \]

       9. associate-/r/ 87.6%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{1}{\frac{\frac{4}{w}}{w \cdot r}}} \cdot r\right) + -4.5 \]

       10. associate-*l/ 87.6%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{1 \cdot r}{\frac{\frac{4}{w}}{w \cdot r}}}\right) + -4.5 \]

       11. *-lft-identity 87.6%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{r}}{\frac{\frac{4}{w}}{w \cdot r}}\right) + -4.5 \]

       12. associate-/r/ 99.9%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{r}{\frac{4}{w}} \cdot \left(w \cdot r\right)}\right) + -4.5 \]

       13. associate-/r/ 99.9%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{r}{4} \cdot w\right)} \cdot \left(w \cdot r\right)\right) + -4.5 \]

       14. associate-*r* 87.6%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) + -4.5 \]

       15. *-commutative 87.6%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{4}}\right) + -4.5 \]

       16. associate-*l* 99.9%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{w \cdot \left(\left(w \cdot r\right) \cdot \frac{r}{4}\right)}\right) + -4.5 \]

   12. Simplified 99.9%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{w \cdot \left(\left(w \cdot r\right) \cdot \frac{r}{4}\right)}\right) + -4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 10^{+183}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{4}\right)\right)\\ \end{array} \]

Alternative 4: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1 \cdot 10^{+26} \lor \neg \left(v \leq 2.9 \cdot 10^{-107}\right):\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1e+26) (not (<= v 2.9e-107)))
     (+ -4.5 (- (+ 3.0 t_0) (* w (* (* r w) (/ r 4.0)))))
     (+ t_0 (- -1.5 (* (* w (* r w)) (/ r 2.6666666666666665)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1e+26) || !(v <= 2.9e-107)) {
		tmp = -4.5 + ((3.0 + t_0) - (w * ((r * w) * (r / 4.0))));
	} else {
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-1d+26)) .or. (.not. (v <= 2.9d-107))) then
        tmp = (-4.5d0) + ((3.0d0 + t_0) - (w * ((r * w) * (r / 4.0d0))))
    else
        tmp = t_0 + ((-1.5d0) - ((w * (r * w)) * (r / 2.6666666666666665d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1e+26) || !(v <= 2.9e-107)) {
		tmp = -4.5 + ((3.0 + t_0) - (w * ((r * w) * (r / 4.0))));
	} else {
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1e+26) or not (v <= 2.9e-107):
		tmp = -4.5 + ((3.0 + t_0) - (w * ((r * w) * (r / 4.0))))
	else:
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1e+26) || !(v <= 2.9e-107))
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(w * Float64(Float64(r * w) * Float64(r / 4.0)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(w * Float64(r * w)) * Float64(r / 2.6666666666666665))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -1e+26) || ~((v <= 2.9e-107)))
		tmp = -4.5 + ((3.0 + t_0) - (w * ((r * w) * (r / 4.0))));
	else
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1e+26], N[Not[LessEqual[v, 2.9e-107]], $MachinePrecision]], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(w * N[(N[(r * w), $MachinePrecision] * N[(r / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(r / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.00000000000000005e26 or 2.8999999999999998e-107 < v

   1. Initial program 82.5%

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

   3. Simplified 77.5%

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

      1. *-un-lft-identity 77.5%

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

      2. add-sqr-sqrt 77.5%

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

      3. times-frac 77.5%

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

      4. unswap-sqr 77.5%

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

      5. sqrt-prod 44.6%

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

      6. add-sqr-sqrt 60.9%

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

      7. unswap-sqr 79.3%

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

      8. sqrt-prod 57.4%

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

      9. add-sqr-sqrt 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. times-frac 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. associate-/r/ 98.5%

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

      2. metadata-eval 98.5%

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

      3. *-commutative 98.5%

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

      4. associate-/r* 94.3%

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

   9. Simplified 94.3%

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

   10. Taylor expanded in v around inf 78.2%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)}\right) + -4.5 \]
   11. Step-by-step derivation

       1. unpow2 78.2%

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

       2. associate-*r* 78.2%

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

       3. unpow2 78.2%

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

       4. associate-*r* 86.9%

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

       5. associate-*r* 86.9%

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

       6. metadata-eval 86.9%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\left(\color{blue}{\frac{1}{4}} \cdot w\right) \cdot w\right) \cdot r\right) \cdot r\right) + -4.5 \]

       7. associate-/r/ 86.9%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\color{blue}{\frac{1}{\frac{4}{w}}} \cdot w\right) \cdot r\right) \cdot r\right) + -4.5 \]

       8. associate-*r* 95.4%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{1}{\frac{4}{w}} \cdot \left(w \cdot r\right)\right)} \cdot r\right) + -4.5 \]

       9. associate-/r/ 95.3%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{1}{\frac{\frac{4}{w}}{w \cdot r}}} \cdot r\right) + -4.5 \]

       10. associate-*l/ 95.3%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{1 \cdot r}{\frac{\frac{4}{w}}{w \cdot r}}}\right) + -4.5 \]

       11. *-lft-identity 95.3%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{r}}{\frac{\frac{4}{w}}{w \cdot r}}\right) + -4.5 \]

       12. associate-/r/ 99.3%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{r}{\frac{4}{w}} \cdot \left(w \cdot r\right)}\right) + -4.5 \]

       13. associate-/r/ 99.3%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{r}{4} \cdot w\right)} \cdot \left(w \cdot r\right)\right) + -4.5 \]

       14. associate-*r* 95.3%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) + -4.5 \]

       15. *-commutative 95.3%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{4}}\right) + -4.5 \]

       16. associate-*l* 98.6%

           \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{w \cdot \left(\left(w \cdot r\right) \cdot \frac{r}{4}\right)}\right) + -4.5 \]

   12. Simplified 98.6%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{w \cdot \left(\left(w \cdot r\right) \cdot \frac{r}{4}\right)}\right) + -4.5 \]

   if -1.00000000000000005e26 < v < 2.8999999999999998e-107

   1. Initial program 85.6%

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

      1. associate--l- 85.6%

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

      2. +-commutative 85.6%

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

      3. associate--l+ 85.6%

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

      4. +-commutative 85.6%

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

      5. associate--r+ 85.6%

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

      6. metadata-eval 85.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*r* 86.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}}{1 - v}\right) \]

      8. *-commutative 86.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) \]

      9. associate-/l* 86.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)}}}\right) \]

      10. *-commutative 86.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   3. Simplified 86.4%

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

   4. Taylor expanded in v around 0 86.4%

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

      1. unpow2 86.4%

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

      2. associate-*l* 98.1%

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

   6. Simplified 98.1%

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

      1. associate-/r/ 98.2%

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

   8. Applied egg-rr 98.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+26} \lor \neg \left(v \leq 2.9 \cdot 10^{-107}\right):\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \end{array} \]

Alternative 5: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 5 \cdot 10^{-48}:\\ \;\;\;\;t_0 + \left(-1.5 - w \cdot \left(r \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)\right)\\ \mathbf{elif}\;r \leq 8.5 \cdot 10^{+80}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 5e-48)
     (+ t_0 (- -1.5 (* w (* r (* r (* w 0.375))))))
     (if (<= r 8.5e+80)
       (+ t_0 (- -1.5 (* (* (* r r) (* w w)) 0.25)))
       (+ t_0 (- -1.5 (* (* w (* r w)) (/ r 2.6666666666666665))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 5e-48) {
		tmp = t_0 + (-1.5 - (w * (r * (r * (w * 0.375)))));
	} else if (r <= 8.5e+80) {
		tmp = t_0 + (-1.5 - (((r * r) * (w * w)) * 0.25));
	} else {
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 5d-48) then
        tmp = t_0 + ((-1.5d0) - (w * (r * (r * (w * 0.375d0)))))
    else if (r <= 8.5d+80) then
        tmp = t_0 + ((-1.5d0) - (((r * r) * (w * w)) * 0.25d0))
    else
        tmp = t_0 + ((-1.5d0) - ((w * (r * w)) * (r / 2.6666666666666665d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 5e-48) {
		tmp = t_0 + (-1.5 - (w * (r * (r * (w * 0.375)))));
	} else if (r <= 8.5e+80) {
		tmp = t_0 + (-1.5 - (((r * r) * (w * w)) * 0.25));
	} else {
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 5e-48:
		tmp = t_0 + (-1.5 - (w * (r * (r * (w * 0.375)))))
	elif r <= 8.5e+80:
		tmp = t_0 + (-1.5 - (((r * r) * (w * w)) * 0.25))
	else:
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 5e-48)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(w * Float64(r * Float64(r * Float64(w * 0.375))))));
	elseif (r <= 8.5e+80)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * r) * Float64(w * w)) * 0.25)));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(w * Float64(r * w)) * Float64(r / 2.6666666666666665))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 5e-48)
		tmp = t_0 + (-1.5 - (w * (r * (r * (w * 0.375)))));
	elseif (r <= 8.5e+80)
		tmp = t_0 + (-1.5 - (((r * r) * (w * w)) * 0.25));
	else
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 5e-48], N[(t$95$0 + N[(-1.5 - N[(w * N[(r * N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 8.5e+80], N[(t$95$0 + N[(-1.5 - N[(N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(r / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 4.9999999999999999e-48

   1. Initial program 81.9%

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

      1. associate--l- 81.9%

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

      2. +-commutative 81.9%

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

      3. associate--l+ 81.9%

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

      4. +-commutative 81.9%

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

      5. associate--r+ 81.9%

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

      6. metadata-eval 81.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*r* 81.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}}{1 - v}\right) \]

      8. *-commutative 81.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) \]

      9. associate-/l* 82.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)}}}\right) \]

      10. *-commutative 82.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   3. Simplified 83.0%

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

   4. Taylor expanded in v around 0 81.8%

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

      1. unpow2 81.8%

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

      2. associate-*l* 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in w around 0 81.8%

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

      1. unpow2 81.8%

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

      2. associate-*r* 91.7%

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

      3. associate-/r* 91.7%

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

      4. *-commutative 91.7%

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

      5. associate-/r* 91.7%

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

   9. Simplified 91.7%

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

       1. associate-/r/ 94.1%

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

       2. div-inv 94.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \frac{1}{\frac{\frac{2.6666666666666665}{w}}{r}}\right)} \cdot w\right) \]

       3. clear-num 94.1%

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

       4. div-inv 94.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \frac{1}{\frac{2.6666666666666665}{w}}\right)}\right) \cdot w\right) \]

       5. clear-num 94.1%

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

       6. div-inv 94.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \color{blue}{\left(w \cdot \frac{1}{2.6666666666666665}\right)}\right)\right) \cdot w\right) \]

       7. metadata-eval 94.1%

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

   11. Applied egg-rr 94.1%

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

   if 4.9999999999999999e-48 < r < 8.50000000000000007e80

   1. Initial program 97.1%

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

      1. associate--l- 97.1%

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

      2. +-commutative 97.1%

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

      3. associate--l+ 97.1%

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

      4. +-commutative 97.1%

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

      5. associate--r+ 97.1%

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

      6. metadata-eval 97.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 99.9%

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

   4. Taylor expanded in r around 0 99.9%

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

      1. unpow2 99.9%

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

      2. associate-*l* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in v around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. unpow2 99.9%

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

   9. Simplified 99.9%

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

   if 8.50000000000000007e80 < r

   1. Initial program 82.1%

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

      1. associate--l- 82.1%

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

      2. +-commutative 82.1%

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

      3. associate--l+ 82.1%

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

      4. +-commutative 82.1%

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

      5. associate--r+ 82.1%

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

      6. metadata-eval 82.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*r* 83.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}}{1 - v}\right) \]

      8. *-commutative 83.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) \]

      9. associate-/l* 85.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)}}}\right) \]

      10. *-commutative 85.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   3. Simplified 85.3%

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

   4. Taylor expanded in v around 0 82.4%

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

      1. unpow2 82.4%

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

      2. associate-*l* 90.6%

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

   6. Simplified 90.6%

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

      1. associate-/r/ 90.7%

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

   8. Applied egg-rr 90.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - w \cdot \left(r \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)\right)\\ \mathbf{elif}\;r \leq 8.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \end{array} \]

Alternative 6: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := w \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq 2 \cdot 10^{-90}:\\ \;\;\;\;t_0 + \left(-1.5 - t_1 \cdot \frac{r}{2.6666666666666665}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - t_1 \cdot \frac{r}{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* w (* r w))))
   (if (<= v 2e-90)
     (+ t_0 (- -1.5 (* t_1 (/ r 2.6666666666666665))))
     (+ t_0 (- -1.5 (* t_1 (/ r 4.0)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = w * (r * w);
	double tmp;
	if (v <= 2e-90) {
		tmp = t_0 + (-1.5 - (t_1 * (r / 2.6666666666666665)));
	} else {
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = w * (r * w)
    if (v <= 2d-90) then
        tmp = t_0 + ((-1.5d0) - (t_1 * (r / 2.6666666666666665d0)))
    else
        tmp = t_0 + ((-1.5d0) - (t_1 * (r / 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = w * (r * w);
	double tmp;
	if (v <= 2e-90) {
		tmp = t_0 + (-1.5 - (t_1 * (r / 2.6666666666666665)));
	} else {
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = w * (r * w)
	tmp = 0
	if v <= 2e-90:
		tmp = t_0 + (-1.5 - (t_1 * (r / 2.6666666666666665)))
	else:
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(w * Float64(r * w))
	tmp = 0.0
	if (v <= 2e-90)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * Float64(r / 2.6666666666666665))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * Float64(r / 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = w * (r * w);
	tmp = 0.0;
	if (v <= 2e-90)
		tmp = t_0 + (-1.5 - (t_1 * (r / 2.6666666666666665)));
	else
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 2e-90], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * N[(r / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * N[(r / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < 1.99999999999999999e-90

   1. Initial program 83.1%

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

      1. associate--l- 83.1%

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

      2. +-commutative 83.1%

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

      3. associate--l+ 83.1%

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

      4. +-commutative 83.1%

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

      5. associate--r+ 83.1%

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

      6. metadata-eval 83.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*r* 83.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}}{1 - v}\right) \]

      8. *-commutative 83.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) \]

      9. associate-/l* 84.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)}}}\right) \]

      10. *-commutative 84.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   3. Simplified 84.7%

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

   4. Taylor expanded in v around 0 84.6%

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

      1. unpow2 84.6%

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

      2. associate-*l* 94.1%

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

   6. Simplified 94.1%

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

      1. associate-/r/ 94.2%

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

   8. Applied egg-rr 94.2%

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

   if 1.99999999999999999e-90 < v

   1. Initial program 85.1%

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

      1. associate--l- 85.1%

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

      2. +-commutative 85.1%

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

      3. associate--l+ 85.1%

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

      4. +-commutative 85.1%

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

      5. associate--r+ 85.1%

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

      6. metadata-eval 85.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*r* 84.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}}{1 - v}\right) \]

      8. *-commutative 84.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) \]

      9. associate-/l* 86.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)}}}\right) \]

      10. *-commutative 86.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   3. Simplified 86.1%

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

   4. Taylor expanded in v around inf 87.2%

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

      1. unpow2 87.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

      2. associate-*l* 97.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

   6. Simplified 97.6%

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

      1. associate-/r/ 97.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   8. Applied egg-rr 97.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 2 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{4}\right)\\ \end{array} \]

Alternative 7: 91.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. associate--l- 83.8%

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

   2. +-commutative 83.8%

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

   3. associate--l+ 83.8%

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

   4. +-commutative 83.8%

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

   5. associate--r+ 83.8%

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

   6. metadata-eval 83.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

   7. associate-*r* 83.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}}{1 - v}\right) \]

   8. *-commutative 83.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) \]

   9. associate-/l* 84.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)}}}\right) \]

   10. *-commutative 84.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{1 - v}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

3. Simplified 85.2%

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

4. Taylor expanded in v around 0 83.5%

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

   1. unpow2 83.5%

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

   2. associate-*l* 91.9%

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

6. Simplified 91.9%

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

7. Taylor expanded in w around 0 83.5%

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

   1. unpow2 83.5%

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

   2. associate-*r* 91.9%

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

   3. associate-/r* 91.9%

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

   4. *-commutative 91.9%

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

   5. associate-/r* 91.9%

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

9. Simplified 91.9%

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

    1. associate-/r/ 92.0%

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

    2. div-inv 92.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \frac{1}{\frac{\frac{2.6666666666666665}{w}}{r}}\right)} \cdot w\right) \]

    3. clear-num 92.0%

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

    4. div-inv 92.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \frac{1}{\frac{2.6666666666666665}{w}}\right)}\right) \cdot w\right) \]

    5. clear-num 92.0%

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

    6. div-inv 92.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \color{blue}{\left(w \cdot \frac{1}{2.6666666666666665}\right)}\right)\right) \cdot w\right) \]

    7. metadata-eval 92.0%

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

11. Applied egg-rr 92.0%

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

12. Final simplification 92.0%

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

Alternative 8: 65.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.0067:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 0.0067) (+ (/ 2.0 (* r r)) -1.5) (* (* r r) (* (* w w) -0.375))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.0067) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * r) * ((w * 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) :: tmp
    if (r <= 0.0067d0) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (r * r) * ((w * w) * (-0.375d0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.0067) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * r) * ((w * w) * -0.375);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.0067:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (r * r) * ((w * w) * -0.375)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.0067)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(r * r) * Float64(Float64(w * w) * -0.375));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.0067)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (r * r) * ((w * w) * -0.375);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.0067], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 0.00670000000000000023

   1. Initial program 83.2%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in v around 0 74.6%

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

      1. *-commutative 74.6%

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

      2. unpow2 74.6%

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

      3. unpow2 74.6%

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

   6. Simplified 74.6%

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

   7. Taylor expanded in r around 0 68.3%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   8. Step-by-step derivation

      1. sub-neg 68.3%

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

      2. associate-*r/ 68.3%

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

      3. metadata-eval 68.3%

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

      4. unpow2 68.3%

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

      5. metadata-eval 68.3%

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   9. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + -1.5} \]

   if 0.00670000000000000023 < r

   1. Initial program 85.5%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in v around 0 76.1%

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

      1. *-commutative 76.1%

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

      2. unpow2 76.1%

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

      3. unpow2 76.1%

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

   6. Simplified 76.1%

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

   7. Taylor expanded in r around inf 63.3%

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

      1. associate-*r* 64.4%

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

      2. unpow2 64.4%

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

      3. unpow2 64.4%

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

   9. Simplified 64.4%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.0067:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ \end{array} \]

Alternative 9: 67.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.00105:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 0.00105) (+ (/ 2.0 (* r r)) -1.5) (* (* r w) (* w (* r -0.375)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00105) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * w) * (w * (r * -0.375));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.00105d0) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (r * w) * (w * (r * (-0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00105) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * w) * (w * (r * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.00105:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (r * w) * (w * (r * -0.375))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.00105)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(r * w) * Float64(w * Float64(r * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.00105)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (r * w) * (w * (r * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.00105], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 0.00104999999999999994

   1. Initial program 83.2%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in v around 0 74.6%

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

      1. *-commutative 74.6%

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

      2. unpow2 74.6%

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

      3. unpow2 74.6%

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

   6. Simplified 74.6%

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

   7. Taylor expanded in r around 0 68.3%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   8. Step-by-step derivation

      1. sub-neg 68.3%

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

      2. associate-*r/ 68.3%

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

      3. metadata-eval 68.3%

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

      4. unpow2 68.3%

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

      5. metadata-eval 68.3%

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   9. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + -1.5} \]

   if 0.00104999999999999994 < r

   1. Initial program 85.5%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in v around 0 76.1%

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

      1. *-commutative 76.1%

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

      2. unpow2 76.1%

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

      3. unpow2 76.1%

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

   6. Simplified 76.1%

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

   7. Taylor expanded in r around inf 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l* 64.5%

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

      3. unpow2 64.5%

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

      4. unpow2 64.5%

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

   9. Simplified 64.5%

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

   10. Taylor expanded in w around 0 63.3%

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

       1. *-commutative 63.3%

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

       2. unpow2 63.3%

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

       3. associate-*r* 64.5%

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

       4. unpow2 64.5%

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

       5. associate-*r* 64.7%

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

       6. associate-*r* 69.3%

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

       7. associate-*r* 76.1%

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

       8. *-commutative 76.1%

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

       9. associate-*l* 76.1%

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

   12. Simplified 76.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.00105:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\\ \end{array} \]

Alternative 10: 57.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -1.5 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Simplified 76.9%

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

4. Taylor expanded in v around 0 75.0%

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

   1. *-commutative 75.0%

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

   2. unpow2 75.0%

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

   3. unpow2 75.0%

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

6. Simplified 75.0%

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

7. Taylor expanded in r around 0 55.4%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
8. Step-by-step derivation

   1. sub-neg 55.4%

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

   2. associate-*r/ 55.4%

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

   3. metadata-eval 55.4%

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

   4. unpow2 55.4%

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

   5. metadata-eval 55.4%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

9. Simplified 55.4%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + -1.5} \]

10. Final simplification 55.4%

    \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

Alternative 11: 44.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (/ 2.0 (* r r)))

C

double code(double v, double w, double r) {
	return 2.0 / (r * 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)
end function

Java

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

Python

def code(v, w, r):
	return 2.0 / (r * r)

Julia

function code(v, w, r)
	return Float64(2.0 / Float64(r * r))
end

MATLAB

function tmp = code(v, w, r)
	tmp = 2.0 / (r * r);
end

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Simplified 76.9%

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

4. Taylor expanded in v around 0 75.0%

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

   1. *-commutative 75.0%

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

   2. unpow2 75.0%

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

   3. unpow2 75.0%

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

6. Simplified 75.0%

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

7. Taylor expanded in r around 0 39.4%

   \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 39.4%

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

9. Simplified 39.4%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]

10. Final simplification 39.4%

    \[\leadsto \frac{2}{r \cdot r} \]

Reproduce

herbie shell --seed 2023275 
(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))