Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 99.1%

Time: 11.0s

Alternatives: 16

Speedup: 1.4×

• 52.3% 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: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+298}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e+298)
     (+
      t_0
      (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* w (* r w))))))
     (+ t_0 (- -1.5 (* 0.375 (* (* r w) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+298) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((w * w) <= 2d+298) then
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (w * (r * w)))))
    else
        tmp = t_0 + ((-1.5d0) - (0.375d0 * ((r * w) * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+298) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 2e+298:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))))
	else:
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e+298)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(w * Float64(r * w))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 2e+298)
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	else
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e+298], N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.9999999999999999e298

   1. Initial program 92.5%

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

      1. associate--l- 92.5%

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

      2. +-commutative 92.5%

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

      3. associate--l+ 92.5%

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

      4. +-commutative 92.5%

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

      5. associate--r+ 92.5%

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

      6. metadata-eval 92.5%

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

      7. associate-*l/ 95.7%

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

      8. *-commutative 95.7%

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

      9. *-commutative 95.7%

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

      10. *-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in r around 0 95.7%

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

      1. unpow2 95.7%

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

      2. *-commutative 95.7%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if 1.9999999999999999e298 < (*.f64 w w)

   1. Initial program 61.6%

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

      1. associate--l- 61.6%

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

      2. +-commutative 61.6%

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

      3. associate--l+ 61.6%

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

      4. +-commutative 61.6%

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

      5. associate--r+ 61.6%

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

      6. metadata-eval 61.6%

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

      7. associate-*l/ 61.6%

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

      8. *-commutative 61.6%

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

      9. *-commutative 61.6%

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

      10. *-commutative 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in r around 0 61.6%

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

      1. unpow2 61.6%

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

      2. *-commutative 61.6%

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

      3. unpow2 61.6%

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

      4. swap-sqr 99.9%

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

      5. unpow2 99.9%

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

      6. *-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. unpow2 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in v around 0 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate--l- 86.0%

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

   2. +-commutative 86.0%

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

   3. associate--l+ 86.0%

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

   4. +-commutative 86.0%

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

   5. associate--r+ 86.0%

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

   6. metadata-eval 86.0%

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

   7. associate-*l/ 88.5%

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

   8. *-commutative 88.5%

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

   9. *-commutative 88.5%

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

   10. *-commutative 88.5%

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

3. Simplified 88.5%

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

4. Taylor expanded in r around 0 80.1%

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

   1. unpow2 80.1%

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

   2. *-commutative 80.1%

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

   3. unpow2 80.1%

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

   4. swap-sqr 99.8%

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

   5. unpow2 99.8%

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

   6. *-commutative 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 0:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{elif}\;w \cdot w \leq 2 \cdot 10^{+298}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 0.0)
     (+ t_0 (- -1.5 (* 0.375 (* r (* w (* r w))))))
     (if (<= (* w w) 2e+298)
       (+
        t_0
        (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* r (* w w))))))
       (+ t_0 (- -1.5 (* 0.375 (* (* r w) (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 0.0) {
		tmp = t_0 + (-1.5 - (0.375 * (r * (w * (r * w)))));
	} else if ((w * w) <= 2e+298) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((w * w) <= 0.0d0) then
        tmp = t_0 + ((-1.5d0) - (0.375d0 * (r * (w * (r * w)))))
    else if ((w * w) <= 2d+298) then
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (r * (w * w)))))
    else
        tmp = t_0 + ((-1.5d0) - (0.375d0 * ((r * w) * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 0.0) {
		tmp = t_0 + (-1.5 - (0.375 * (r * (w * (r * w)))));
	} else if ((w * w) <= 2e+298) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 0.0:
		tmp = t_0 + (-1.5 - (0.375 * (r * (w * (r * w)))))
	elif (w * w) <= 2e+298:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	else:
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 0.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(w * Float64(r * w))))));
	elseif (Float64(w * w) <= 2e+298)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 0.0)
		tmp = t_0 + (-1.5 - (0.375 * (r * (w * (r * w)))));
	elseif ((w * w) <= 2e+298)
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	else
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 0.0], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(w * w), $MachinePrecision], 2e+298], N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 w w) < 0.0

   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/ 89.6%

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

      8. *-commutative 89.6%

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

      9. *-commutative 89.6%

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

      10. *-commutative 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in r around 0 89.6%

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

      1. unpow2 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in v around 0 96.6%

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

   if 0.0 < (*.f64 w w) < 1.9999999999999999e298

   1. Initial program 94.2%

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

      1. associate--l- 94.2%

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

      2. +-commutative 94.2%

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

      3. associate--l+ 94.2%

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

      4. +-commutative 94.2%

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

      5. associate--r+ 94.2%

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

      6. metadata-eval 94.2%

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

      7. associate-*l/ 99.2%

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

      8. *-commutative 99.2%

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

      9. *-commutative 99.2%

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

      10. *-commutative 99.2%

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

   3. Simplified 99.2%

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

   if 1.9999999999999999e298 < (*.f64 w w)

   1. Initial program 61.6%

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

      1. associate--l- 61.6%

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

      2. +-commutative 61.6%

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

      3. associate--l+ 61.6%

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

      4. +-commutative 61.6%

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

      5. associate--r+ 61.6%

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

      6. metadata-eval 61.6%

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

      7. associate-*l/ 61.6%

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

      8. *-commutative 61.6%

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

      9. *-commutative 61.6%

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

      10. *-commutative 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in r around 0 61.6%

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

      1. unpow2 61.6%

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

      2. *-commutative 61.6%

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

      3. unpow2 61.6%

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

      4. swap-sqr 99.9%

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

      5. unpow2 99.9%

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

      6. *-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. unpow2 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in v around 0 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 0:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{elif}\;w \cdot w \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := r \cdot \left(w \cdot \left(r \cdot w\right)\right)\\ \mathbf{if}\;v \leq -1.5 \cdot 10^{+29}:\\ \;\;\;\;t_0 + \left(-1.5 - t_1 \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 4.2 \cdot 10^{-13}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + t_1 \cdot \left(\frac{0.125}{v} - 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* r (* w (* r w)))))
   (if (<= v -1.5e+29)
     (+ t_0 (- -1.5 (* t_1 0.25)))
     (if (<= v 4.2e-13)
       (+ t_0 (- -1.5 (* 0.375 (* (* r w) (* r w)))))
       (+ t_0 (+ -1.5 (* t_1 (- (/ 0.125 v) 0.25))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r * (w * (r * w));
	double tmp;
	if (v <= -1.5e+29) {
		tmp = t_0 + (-1.5 - (t_1 * 0.25));
	} else if (v <= 4.2e-13) {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	} else {
		tmp = t_0 + (-1.5 + (t_1 * ((0.125 / v) - 0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = r * (w * (r * w))
    if (v <= (-1.5d+29)) then
        tmp = t_0 + ((-1.5d0) - (t_1 * 0.25d0))
    else if (v <= 4.2d-13) then
        tmp = t_0 + ((-1.5d0) - (0.375d0 * ((r * w) * (r * w))))
    else
        tmp = t_0 + ((-1.5d0) + (t_1 * ((0.125d0 / v) - 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r * (w * (r * w));
	double tmp;
	if (v <= -1.5e+29) {
		tmp = t_0 + (-1.5 - (t_1 * 0.25));
	} else if (v <= 4.2e-13) {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	} else {
		tmp = t_0 + (-1.5 + (t_1 * ((0.125 / v) - 0.25)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = r * (w * (r * w))
	tmp = 0
	if v <= -1.5e+29:
		tmp = t_0 + (-1.5 - (t_1 * 0.25))
	elif v <= 4.2e-13:
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))))
	else:
		tmp = t_0 + (-1.5 + (t_1 * ((0.125 / v) - 0.25)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(r * Float64(w * Float64(r * w)))
	tmp = 0.0
	if (v <= -1.5e+29)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * 0.25)));
	elseif (v <= 4.2e-13)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(t_1 * Float64(Float64(0.125 / v) - 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = r * (w * (r * w));
	tmp = 0.0;
	if (v <= -1.5e+29)
		tmp = t_0 + (-1.5 - (t_1 * 0.25));
	elseif (v <= 4.2e-13)
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	else
		tmp = t_0 + (-1.5 + (t_1 * ((0.125 / v) - 0.25)));
	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[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1.5e+29], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 4.2e-13], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(t$95$1 * N[(N[(0.125 / v), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.5e29

   1. Initial program 84.0%

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

      1. associate--l- 84.0%

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

      2. +-commutative 84.0%

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

      3. associate--l+ 84.0%

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

      4. +-commutative 84.0%

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

      5. associate--r+ 84.0%

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

      6. metadata-eval 84.0%

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

      7. associate-*l/ 87.3%

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

      8. *-commutative 87.3%

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

      9. *-commutative 87.3%

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

      10. *-commutative 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in r around 0 87.3%

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

      1. unpow2 87.3%

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

      2. *-commutative 87.3%

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

      3. associate-*r* 98.0%

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

      4. *-commutative 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in v around inf 98.0%

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

   if -1.5e29 < v < 4.19999999999999977e-13

   1. Initial program 86.6%

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

      1. associate--l- 86.6%

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

      2. +-commutative 86.6%

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

      3. associate--l+ 86.6%

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

      4. +-commutative 86.6%

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

      5. associate--r+ 86.6%

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

      6. metadata-eval 86.6%

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

      7. associate-*l/ 86.6%

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

      8. *-commutative 86.6%

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

      9. *-commutative 86.6%

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

      10. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in r around 0 77.4%

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

      1. unpow2 77.4%

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

      2. *-commutative 77.4%

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

      3. unpow2 77.4%

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

      4. swap-sqr 99.9%

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

      5. unpow2 99.9%

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

      6. *-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. unpow2 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in v around 0 99.3%

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

   if 4.19999999999999977e-13 < v

   1. Initial program 86.5%

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

      1. associate--l- 86.5%

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

      2. +-commutative 86.5%

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

      3. associate--l+ 86.5%

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

      4. +-commutative 86.5%

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

      5. associate--r+ 86.5%

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

      6. metadata-eval 86.5%

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

      7. associate-*l/ 93.9%

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

      8. *-commutative 93.9%

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

      9. *-commutative 93.9%

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

      10. *-commutative 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in r around 0 93.9%

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

      1. unpow2 93.9%

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

      2. *-commutative 93.9%

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

      3. associate-*r* 99.7%

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

      4. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in v around inf 96.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot \color{blue}{\left(0.25 - 0.125 \cdot \frac{1}{v}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 96.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot \left(0.25 - \color{blue}{\frac{0.125 \cdot 1}{v}}\right)\right) \]

      2. metadata-eval 96.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot \left(0.25 - \frac{\color{blue}{0.125}}{v}\right)\right) \]

   9. Simplified 96.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot \color{blue}{\left(0.25 - \frac{0.125}{v}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)\right)\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate--l- 86.0%

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

   2. +-commutative 86.0%

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

   3. associate--l+ 86.0%

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

   4. +-commutative 86.0%

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

   5. associate--r+ 86.0%

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

   6. metadata-eval 86.0%

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

   7. associate-*l/ 88.5%

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

   8. *-commutative 88.5%

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

   9. *-commutative 88.5%

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

   10. *-commutative 88.5%

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

3. Simplified 88.5%

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

4. Taylor expanded in r around 0 80.1%

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

   1. unpow2 80.1%

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

   2. *-commutative 80.1%

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

   3. unpow2 80.1%

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

   4. swap-sqr 99.8%

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

   5. unpow2 99.8%

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

   6. *-commutative 99.8%

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

6. Simplified 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 6: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(w \cdot \left(r \cdot w\right)\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.35 \cdot 10^{+29} \lor \neg \left(v \leq 8.5 \cdot 10^{-24}\right):\\ \;\;\;\;t_1 + \left(-1.5 - t_0 \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - 0.375 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* w (* r w)))) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -1.35e+29) (not (<= v 8.5e-24)))
     (+ t_1 (- -1.5 (* t_0 0.25)))
     (+ t_1 (- -1.5 (* 0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = r * (w * (r * w));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.35e+29) || !(v <= 8.5e-24)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_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 = r * (w * (r * w))
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-1.35d+29)) .or. (.not. (v <= 8.5d-24))) then
        tmp = t_1 + ((-1.5d0) - (t_0 * 0.25d0))
    else
        tmp = t_1 + ((-1.5d0) - (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * (w * (r * w));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.35e+29) || !(v <= 8.5e-24)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * (w * (r * w))
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1.35e+29) or not (v <= 8.5e-24):
		tmp = t_1 + (-1.5 - (t_0 * 0.25))
	else:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(w * Float64(r * w)))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1.35e+29) || !(v <= 8.5e-24))
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * 0.25)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * (w * (r * w));
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -1.35e+29) || ~((v <= 8.5e-24)))
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	else
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1.35e+29], N[Not[LessEqual[v, 8.5e-24]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.35e29 or 8.5000000000000002e-24 < 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-*l/ 90.4%

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

      8. *-commutative 90.4%

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

      9. *-commutative 90.4%

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

      10. *-commutative 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in r around 0 90.4%

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

      1. unpow2 90.4%

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

      2. *-commutative 90.4%

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

      3. associate-*r* 98.2%

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

      4. *-commutative 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in v around inf 96.4%

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

   if -1.35e29 < v < 8.5000000000000002e-24

   1. Initial program 86.9%

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

      1. associate--l- 86.9%

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

      2. +-commutative 86.9%

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

      3. associate--l+ 86.9%

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

      4. +-commutative 86.9%

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

      5. associate--r+ 86.9%

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

      6. metadata-eval 86.9%

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

      7. associate-*l/ 86.9%

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

      8. *-commutative 86.9%

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

      9. *-commutative 86.9%

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

      10. *-commutative 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in r around 0 86.9%

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

      1. unpow2 86.9%

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

      2. *-commutative 86.9%

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

      3. associate-*r* 95.6%

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

      4. *-commutative 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in v around 0 95.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.35 \cdot 10^{+29} \lor \neg \left(v \leq 8.5 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -4.5 \cdot 10^{+29} \lor \neg \left(v \leq 4.2 \cdot 10^{-16}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -4.5e+29) (not (<= v 4.2e-16)))
     (+ t_0 (- -1.5 (* (* r (* w (* r w))) 0.25)))
     (+ t_0 (- -1.5 (* 0.375 (* (* r w) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -4.5e+29) || !(v <= 4.2e-16)) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 0.25));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-4.5d+29)) .or. (.not. (v <= 4.2d-16))) then
        tmp = t_0 + ((-1.5d0) - ((r * (w * (r * w))) * 0.25d0))
    else
        tmp = t_0 + ((-1.5d0) - (0.375d0 * ((r * w) * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -4.5e+29) || !(v <= 4.2e-16)) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 0.25));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -4.5e+29) or not (v <= 4.2e-16):
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 0.25))
	else:
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -4.5e+29) || !(v <= 4.2e-16))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * 0.25)));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -4.5e+29) || ~((v <= 4.2e-16)))
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 0.25));
	else
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -4.5e+29], N[Not[LessEqual[v, 4.2e-16]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -4.5000000000000002e29 or 4.2000000000000002e-16 < v

   1. Initial program 85.7%

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

      1. associate--l- 85.7%

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

      2. +-commutative 85.7%

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

      3. associate--l+ 85.7%

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

      4. +-commutative 85.7%

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

      5. associate--r+ 85.7%

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

      6. metadata-eval 85.7%

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

      7. associate-*l/ 91.1%

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

      8. *-commutative 91.1%

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

      9. *-commutative 91.1%

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

      10. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in r around 0 91.1%

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

      1. unpow2 91.1%

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

      2. *-commutative 91.1%

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

      3. associate-*r* 99.0%

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

      4. *-commutative 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in v around inf 97.1%

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

   if -4.5000000000000002e29 < v < 4.2000000000000002e-16

   1. Initial program 86.3%

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

      1. associate--l- 86.3%

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

      2. +-commutative 86.3%

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

      3. associate--l+ 86.3%

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

      4. +-commutative 86.3%

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

      5. associate--r+ 86.3%

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

      6. metadata-eval 86.3%

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

      7. associate-*l/ 86.3%

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

      8. *-commutative 86.3%

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

      9. *-commutative 86.3%

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

      10. *-commutative 86.3%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in r around 0 76.9%

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

      1. unpow2 76.9%

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

      2. *-commutative 76.9%

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

      3. unpow2 76.9%

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

      4. swap-sqr 99.9%

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

      5. unpow2 99.9%

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

      6. *-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. unpow2 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in v around 0 99.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4.5 \cdot 10^{+29} \lor \neg \left(v \leq 4.2 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 8: 63.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 8.4 \cdot 10^{+35} \lor \neg \left(r \leq 1.85 \cdot 10^{+65}\right) \land \left(r \leq 1.75 \cdot 10^{+96} \lor \neg \left(r \leq 6.5 \cdot 10^{+106}\right) \land r \leq 3.9 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{w}{\frac{-4}{\left(r \cdot r\right) \cdot w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= r 8.4e+35)
         (and (not (<= r 1.85e+65))
              (or (<= r 1.75e+96)
                  (and (not (<= r 6.5e+106)) (<= r 3.9e+136)))))
   (+ (/ 2.0 (* r r)) -1.5)
   (/ w (/ -4.0 (* (* r r) w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((r <= 8.4e+35) || (!(r <= 1.85e+65) && ((r <= 1.75e+96) || (!(r <= 6.5e+106) && (r <= 3.9e+136))))) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = w / (-4.0 / ((r * r) * w));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if ((r <= 8.4d+35) .or. (.not. (r <= 1.85d+65)) .and. (r <= 1.75d+96) .or. (.not. (r <= 6.5d+106)) .and. (r <= 3.9d+136)) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = w / ((-4.0d0) / ((r * r) * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((r <= 8.4e+35) || (!(r <= 1.85e+65) && ((r <= 1.75e+96) || (!(r <= 6.5e+106) && (r <= 3.9e+136))))) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = w / (-4.0 / ((r * r) * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (r <= 8.4e+35) or (not (r <= 1.85e+65) and ((r <= 1.75e+96) or (not (r <= 6.5e+106) and (r <= 3.9e+136)))):
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = w / (-4.0 / ((r * r) * w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((r <= 8.4e+35) || (!(r <= 1.85e+65) && ((r <= 1.75e+96) || (!(r <= 6.5e+106) && (r <= 3.9e+136)))))
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(w / Float64(-4.0 / Float64(Float64(r * r) * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((r <= 8.4e+35) || (~((r <= 1.85e+65)) && ((r <= 1.75e+96) || (~((r <= 6.5e+106)) && (r <= 3.9e+136)))))
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = w / (-4.0 / ((r * r) * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[r, 8.4e+35], And[N[Not[LessEqual[r, 1.85e+65]], $MachinePrecision], Or[LessEqual[r, 1.75e+96], And[N[Not[LessEqual[r, 6.5e+106]], $MachinePrecision], LessEqual[r, 3.9e+136]]]]], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(w / N[(-4.0 / N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8.3999999999999996e35 or 1.84999999999999997e65 < r < 1.7499999999999999e96 or 6.5000000000000003e106 < r < 3.90000000000000019e136

   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. sub-neg 85.1%

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

      2. +-commutative 85.1%

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

      3. associate--l+ 85.1%

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

      4. associate-/l* 86.5%

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

      5. distribute-neg-frac 86.5%

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

      6. associate-/r/ 86.5%

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

      7. fma-def 86.5%

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

      8. sub-neg 86.5%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in r around 0 67.0%

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

      1. sub-neg 67.0%

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

      2. associate-*r/ 67.0%

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

      3. metadata-eval 67.0%

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

      4. unpow2 67.0%

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

      5. metadata-eval 67.0%

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

   6. Simplified 67.0%

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

   if 8.3999999999999996e35 < r < 1.84999999999999997e65 or 1.7499999999999999e96 < r < 6.5000000000000003e106 or 3.90000000000000019e136 < r

   1. Initial program 89.3%

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

      1. sub-neg 89.3%

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

      2. +-commutative 89.3%

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

      3. associate--l+ 89.3%

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

      4. associate-/l* 96.3%

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

      5. distribute-neg-frac 96.3%

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

      6. associate-/r/ 96.3%

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

      7. fma-def 96.3%

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

      8. sub-neg 96.3%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in r around inf 65.6%

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

      1. unpow2 65.6%

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

      2. associate-/l* 67.3%

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

      3. *-commutative 67.3%

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

      4. unpow2 67.3%

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

      5. *-commutative 67.3%

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

      6. fma-neg 67.3%

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

      7. metadata-eval 67.3%

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

   6. Simplified 67.3%

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

      1. associate-/l* 67.7%

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

      2. div-inv 67.8%

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

      3. associate-*l* 67.8%

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

   8. Applied egg-rr 67.8%

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

      1. associate-*r/ 67.7%

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

      2. *-rgt-identity 67.7%

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

      3. associate-/l/ 67.7%

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

      4. associate-*r* 67.8%

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

   10. Simplified 67.8%

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

   11. Taylor expanded in v around inf 68.0%

       \[\leadsto \frac{w}{\color{blue}{\frac{-4}{w \cdot {r}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 68.0%

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

   13. Simplified 68.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8.4 \cdot 10^{+35} \lor \neg \left(r \leq 1.85 \cdot 10^{+65}\right) \land \left(r \leq 1.75 \cdot 10^{+96} \lor \neg \left(r \leq 6.5 \cdot 10^{+106}\right) \land r \leq 3.9 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{w}{\frac{-4}{\left(r \cdot r\right) \cdot w}}\\ \end{array} \]

Alternative 9: 64.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 8 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot r\right) \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 8e-140) t_0 (+ t_0 (- -1.5 (* (* r r) (* 0.375 (* w w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 8e-140) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - ((r * r) * (0.375 * (w * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 8d-140) then
        tmp = t_0
    else
        tmp = t_0 + ((-1.5d0) - ((r * r) * (0.375d0 * (w * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 8e-140) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - ((r * r) * (0.375 * (w * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 8e-140:
		tmp = t_0
	else:
		tmp = t_0 + (-1.5 - ((r * r) * (0.375 * (w * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 8e-140)
		tmp = t_0;
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * r) * Float64(0.375 * Float64(w * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 8e-140)
		tmp = t_0;
	else
		tmp = t_0 + (-1.5 - ((r * r) * (0.375 * (w * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 8e-140], t$95$0, N[(t$95$0 + N[(-1.5 - N[(N[(r * r), $MachinePrecision] * N[(0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 7.9999999999999999e-140

   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-*l/ 83.9%

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

      8. *-commutative 83.9%

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

      9. *-commutative 83.9%

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

      10. *-commutative 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in r around 0 78.0%

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

      1. unpow2 78.0%

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

      2. *-commutative 78.0%

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

      3. unpow2 78.0%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.9%

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

      6. *-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around 0 94.3%

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

   10. Taylor expanded in r around 0 52.7%

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

       1. unpow2 52.7%

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

   12. Simplified 52.7%

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

   if 7.9999999999999999e-140 < r

   1. Initial program 92.3%

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

      1. associate--l- 92.3%

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

      2. +-commutative 92.3%

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

      3. associate--l+ 92.3%

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

      4. +-commutative 92.3%

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

      5. associate--r+ 92.3%

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

      6. metadata-eval 92.3%

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

      7. associate-*l/ 96.0%

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

      8. *-commutative 96.0%

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

      9. *-commutative 96.0%

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

      10. *-commutative 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in r around 0 83.3%

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

      1. unpow2 83.3%

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

      2. *-commutative 83.3%

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

      3. unpow2 83.3%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

      6. *-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. add-cbrt-cube 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{0.375 + v \cdot -0.25}{1 - v} \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}}}\right) \]

      2. pow3 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \sqrt[3]{\color{blue}{{\left(\frac{0.375 + v \cdot -0.25}{1 - v}\right)}^{3}}}\right) \]

      3. +-commutative 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \sqrt[3]{{\left(\frac{\color{blue}{v \cdot -0.25 + 0.375}}{1 - v}\right)}^{3}}\right) \]

      4. fma-def 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \sqrt[3]{{\left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v}\right)}^{3}}\right) \]

   10. Applied egg-rr 99.7%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right)}^{3}}}\right) \]

   11. Taylor expanded in v around 0 79.2%

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

       1. associate-*r* 79.2%

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

       2. unpow2 79.2%

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

       3. unpow2 79.2%

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

   13. Simplified 79.2%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot r\right) \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 10: 67.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 8.2 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 8.2e-140) t_0 (+ t_0 (- -1.5 (* (* r (* r (* w w))) 0.25))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 8.2e-140) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - ((r * (r * (w * w))) * 0.25));
	}
	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 <= 8.2d-140) then
        tmp = t_0
    else
        tmp = t_0 + ((-1.5d0) - ((r * (r * (w * w))) * 0.25d0))
    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 <= 8.2e-140) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - ((r * (r * (w * w))) * 0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 8.2e-140:
		tmp = t_0
	else:
		tmp = t_0 + (-1.5 - ((r * (r * (w * w))) * 0.25))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 8.2e-140)
		tmp = t_0;
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(r * Float64(w * w))) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 8.2e-140)
		tmp = t_0;
	else
		tmp = t_0 + (-1.5 - ((r * (r * (w * w))) * 0.25));
	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, 8.2e-140], t$95$0, N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 8.2000000000000003e-140

   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-*l/ 83.9%

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

      8. *-commutative 83.9%

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

      9. *-commutative 83.9%

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

      10. *-commutative 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in r around 0 78.0%

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

      1. unpow2 78.0%

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

      2. *-commutative 78.0%

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

      3. unpow2 78.0%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.9%

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

      6. *-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around 0 94.3%

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

   10. Taylor expanded in r around 0 52.7%

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

       1. unpow2 52.7%

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

   12. Simplified 52.7%

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

   if 8.2000000000000003e-140 < r

   1. Initial program 92.3%

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

      1. associate--l- 92.3%

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

      2. +-commutative 92.3%

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

      3. associate--l+ 92.3%

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

      4. +-commutative 92.3%

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

      5. associate--r+ 92.3%

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

      6. metadata-eval 92.3%

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

      7. associate-*l/ 96.0%

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

      8. *-commutative 96.0%

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

      9. *-commutative 96.0%

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

      10. *-commutative 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in r around 0 96.0%

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

      1. unpow2 96.0%

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

      2. *-commutative 96.0%

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

      3. associate-*r* 98.0%

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

      4. *-commutative 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in v around inf 89.9%

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

   8. Taylor expanded in w around 0 89.6%

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

      1. unpow2 89.6%

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

      2. *-commutative 89.6%

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

   10. Simplified 89.6%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot 0.25\right)\\ \end{array} \]

Alternative 11: 68.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 3.3 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 3.3e-155) t_0 (+ t_0 (- -1.5 (* (* r (* w (* r w))) 0.25))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3.3e-155) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 0.25));
	}
	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 <= 3.3d-155) then
        tmp = t_0
    else
        tmp = t_0 + ((-1.5d0) - ((r * (w * (r * w))) * 0.25d0))
    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 <= 3.3e-155) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 3.3e-155:
		tmp = t_0
	else:
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 0.25))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 3.3e-155)
		tmp = t_0;
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 3.3e-155)
		tmp = t_0;
	else
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 0.25));
	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, 3.3e-155], t$95$0, N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.29999999999999986e-155

   1. Initial program 82.5%

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

      1. associate--l- 82.5%

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

      2. +-commutative 82.5%

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

      3. associate--l+ 82.5%

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

      4. +-commutative 82.5%

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

      5. associate--r+ 82.5%

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

      6. metadata-eval 82.5%

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

      7. associate-*l/ 84.3%

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

      8. *-commutative 84.3%

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

      9. *-commutative 84.3%

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

      10. *-commutative 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in r around 0 78.4%

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

      1. unpow2 78.4%

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

      2. *-commutative 78.4%

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

      3. unpow2 78.4%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

      6. *-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around 0 94.3%

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

   10. Taylor expanded in r around 0 52.1%

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

       1. unpow2 52.1%

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

   12. Simplified 52.1%

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

   if 3.29999999999999986e-155 < r

   1. Initial program 91.4%

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

      1. associate--l- 91.4%

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

      2. +-commutative 91.4%

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

      3. associate--l+ 91.4%

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

      4. +-commutative 91.4%

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

      5. associate--r+ 91.4%

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

      6. metadata-eval 91.4%

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

      7. associate-*l/ 95.1%

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

      8. *-commutative 95.1%

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

      9. *-commutative 95.1%

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

      10. *-commutative 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in r around 0 95.1%

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

      1. unpow2 95.1%

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

      2. *-commutative 95.1%

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

      3. associate-*r* 98.0%

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

      4. *-commutative 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in v around inf 90.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot 0.25\right)\\ \end{array} \]

Alternative 12: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot r\right) \cdot w\\ t_1 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;r \leq 1.42 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 6 \cdot 10^{+106}:\\ \;\;\;\;\frac{w}{\frac{-2.6666666666666665}{t_0}}\\ \mathbf{elif}\;r \leq 7 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{w}{\frac{-4}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r r) w)) (t_1 (+ (/ 2.0 (* r r)) -1.5)))
   (if (<= r 1.42e+35)
     t_1
     (if (<= r 6e+106)
       (/ w (/ -2.6666666666666665 t_0))
       (if (<= r 7e+136) t_1 (/ w (/ -4.0 t_0)))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * r) * w;
	double t_1 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (r <= 1.42e+35) {
		tmp = t_1;
	} else if (r <= 6e+106) {
		tmp = w / (-2.6666666666666665 / t_0);
	} else if (r <= 7e+136) {
		tmp = t_1;
	} else {
		tmp = w / (-4.0 / t_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 = (r * r) * w
    t_1 = (2.0d0 / (r * r)) + (-1.5d0)
    if (r <= 1.42d+35) then
        tmp = t_1
    else if (r <= 6d+106) then
        tmp = w / ((-2.6666666666666665d0) / t_0)
    else if (r <= 7d+136) then
        tmp = t_1
    else
        tmp = w / ((-4.0d0) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * r) * w;
	double t_1 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (r <= 1.42e+35) {
		tmp = t_1;
	} else if (r <= 6e+106) {
		tmp = w / (-2.6666666666666665 / t_0);
	} else if (r <= 7e+136) {
		tmp = t_1;
	} else {
		tmp = w / (-4.0 / t_0);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * r) * w
	t_1 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if r <= 1.42e+35:
		tmp = t_1
	elif r <= 6e+106:
		tmp = w / (-2.6666666666666665 / t_0)
	elif r <= 7e+136:
		tmp = t_1
	else:
		tmp = w / (-4.0 / t_0)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * r) * w)
	t_1 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if (r <= 1.42e+35)
		tmp = t_1;
	elseif (r <= 6e+106)
		tmp = Float64(w / Float64(-2.6666666666666665 / t_0));
	elseif (r <= 7e+136)
		tmp = t_1;
	else
		tmp = Float64(w / Float64(-4.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * r) * w;
	t_1 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if (r <= 1.42e+35)
		tmp = t_1;
	elseif (r <= 6e+106)
		tmp = w / (-2.6666666666666665 / t_0);
	elseif (r <= 7e+136)
		tmp = t_1;
	else
		tmp = w / (-4.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[LessEqual[r, 1.42e+35], t$95$1, If[LessEqual[r, 6e+106], N[(w / N[(-2.6666666666666665 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 7e+136], t$95$1, N[(w / N[(-4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.41999999999999991e35 or 6.0000000000000001e106 < r < 7.00000000000000002e136

   1. Initial program 84.6%

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

      1. sub-neg 84.6%

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

      2. +-commutative 84.6%

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

      3. associate--l+ 84.6%

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

      4. associate-/l* 86.0%

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

      5. distribute-neg-frac 86.0%

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

      6. associate-/r/ 86.0%

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

      7. fma-def 86.0%

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

      8. sub-neg 86.0%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in r around 0 66.8%

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

      1. sub-neg 66.8%

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

      2. associate-*r/ 66.8%

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

      3. metadata-eval 66.8%

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

      4. unpow2 66.8%

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

      5. metadata-eval 66.8%

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

   6. Simplified 66.8%

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

   if 1.41999999999999991e35 < r < 6.0000000000000001e106

   1. Initial program 89.2%

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

      1. sub-neg 89.2%

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

      2. +-commutative 89.2%

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

      3. associate--l+ 89.2%

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

      4. associate-/l* 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. associate-/r/ 100.0%

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

      7. fma-def 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in r around inf 55.0%

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

      1. unpow2 55.0%

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

      2. associate-/l* 60.3%

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

      3. *-commutative 60.3%

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

      4. unpow2 60.3%

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

      5. *-commutative 60.3%

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

      6. fma-neg 60.3%

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

      7. metadata-eval 60.3%

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

   6. Simplified 60.3%

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

      1. associate-/l* 60.1%

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

      2. div-inv 60.3%

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

      3. associate-*l* 60.3%

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

   8. Applied egg-rr 60.3%

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

      1. associate-*r/ 60.0%

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

      2. *-rgt-identity 60.0%

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

      3. associate-/l/ 60.1%

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

      4. associate-*r* 60.2%

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

   10. Simplified 60.2%

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

   11. Taylor expanded in v around 0 56.7%

       \[\leadsto \frac{w}{\color{blue}{\frac{-2.6666666666666665}{w \cdot {r}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 56.7%

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

   13. Simplified 56.7%

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

   if 7.00000000000000002e136 < r

   1. Initial program 91.1%

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

      1. sub-neg 91.1%

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

      2. +-commutative 91.1%

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

      3. associate--l+ 91.1%

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

      4. associate-/l* 95.5%

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

      5. distribute-neg-frac 95.5%

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

      6. associate-/r/ 95.4%

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

      7. fma-def 95.4%

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

      8. sub-neg 95.4%

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

   3. Simplified 66.4%

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

   4. Taylor expanded in r around inf 64.2%

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

      1. unpow2 64.2%

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

      2. associate-/l* 64.2%

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

      3. *-commutative 64.2%

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

      4. unpow2 64.2%

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

      5. *-commutative 64.2%

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

      6. fma-neg 64.2%

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

      7. metadata-eval 64.2%

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

   6. Simplified 64.2%

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

      1. associate-/l* 64.7%

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

      2. div-inv 64.7%

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

      3. associate-*l* 64.7%

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

   8. Applied egg-rr 64.7%

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

      1. associate-*r/ 64.7%

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

      2. *-rgt-identity 64.7%

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

      3. associate-/l/ 64.7%

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

      4. associate-*r* 64.7%

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

   10. Simplified 64.7%

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

   11. Taylor expanded in v around inf 64.7%

       \[\leadsto \frac{w}{\color{blue}{\frac{-4}{w \cdot {r}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 64.7%

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

   13. Simplified 64.7%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.42 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 6 \cdot 10^{+106}:\\ \;\;\;\;\frac{w}{\frac{-2.6666666666666665}{\left(r \cdot r\right) \cdot w}}\\ \mathbf{elif}\;r \leq 7 \cdot 10^{+136}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{w}{\frac{-4}{\left(r \cdot r\right) \cdot w}}\\ \end{array} \]

Alternative 13: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;r \leq 3.15 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 6.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{w}{\frac{-2.6666666666666665}{\left(r \cdot r\right) \cdot w}}\\ \mathbf{elif}\;r \leq 3.35 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{w}{\frac{\frac{\frac{-4}{r}}{r}}{w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5)))
   (if (<= r 3.15e+35)
     t_0
     (if (<= r 6.3e+106)
       (/ w (/ -2.6666666666666665 (* (* r r) w)))
       (if (<= r 3.35e+136) t_0 (/ w (/ (/ (/ -4.0 r) r) w)))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (r <= 3.15e+35) {
		tmp = t_0;
	} else if (r <= 6.3e+106) {
		tmp = w / (-2.6666666666666665 / ((r * r) * w));
	} else if (r <= 3.35e+136) {
		tmp = t_0;
	} else {
		tmp = w / (((-4.0 / r) / r) / w);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / (r * r)) + (-1.5d0)
    if (r <= 3.15d+35) then
        tmp = t_0
    else if (r <= 6.3d+106) then
        tmp = w / ((-2.6666666666666665d0) / ((r * r) * w))
    else if (r <= 3.35d+136) then
        tmp = t_0
    else
        tmp = w / ((((-4.0d0) / r) / r) / w)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (r <= 3.15e+35) {
		tmp = t_0;
	} else if (r <= 6.3e+106) {
		tmp = w / (-2.6666666666666665 / ((r * r) * w));
	} else if (r <= 3.35e+136) {
		tmp = t_0;
	} else {
		tmp = w / (((-4.0 / r) / r) / w);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if r <= 3.15e+35:
		tmp = t_0
	elif r <= 6.3e+106:
		tmp = w / (-2.6666666666666665 / ((r * r) * w))
	elif r <= 3.35e+136:
		tmp = t_0
	else:
		tmp = w / (((-4.0 / r) / r) / w)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if (r <= 3.15e+35)
		tmp = t_0;
	elseif (r <= 6.3e+106)
		tmp = Float64(w / Float64(-2.6666666666666665 / Float64(Float64(r * r) * w)));
	elseif (r <= 3.35e+136)
		tmp = t_0;
	else
		tmp = Float64(w / Float64(Float64(Float64(-4.0 / r) / r) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if (r <= 3.15e+35)
		tmp = t_0;
	elseif (r <= 6.3e+106)
		tmp = w / (-2.6666666666666665 / ((r * r) * w));
	elseif (r <= 3.35e+136)
		tmp = t_0;
	else
		tmp = w / (((-4.0 / r) / r) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[LessEqual[r, 3.15e+35], t$95$0, If[LessEqual[r, 6.3e+106], N[(w / N[(-2.6666666666666665 / N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 3.35e+136], t$95$0, N[(w / N[(N[(N[(-4.0 / r), $MachinePrecision] / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 3.14999999999999985e35 or 6.29999999999999974e106 < r < 3.35e136

   1. Initial program 84.6%

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

      1. sub-neg 84.6%

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

      2. +-commutative 84.6%

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

      3. associate--l+ 84.6%

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

      4. associate-/l* 86.0%

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

      5. distribute-neg-frac 86.0%

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

      6. associate-/r/ 86.0%

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

      7. fma-def 86.0%

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

      8. sub-neg 86.0%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in r around 0 66.8%

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

      1. sub-neg 66.8%

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

      2. associate-*r/ 66.8%

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

      3. metadata-eval 66.8%

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

      4. unpow2 66.8%

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

      5. metadata-eval 66.8%

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

   6. Simplified 66.8%

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

   if 3.14999999999999985e35 < r < 6.29999999999999974e106

   1. Initial program 89.2%

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

      1. sub-neg 89.2%

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

      2. +-commutative 89.2%

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

      3. associate--l+ 89.2%

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

      4. associate-/l* 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. associate-/r/ 100.0%

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

      7. fma-def 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in r around inf 55.0%

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

      1. unpow2 55.0%

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

      2. associate-/l* 60.3%

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

      3. *-commutative 60.3%

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

      4. unpow2 60.3%

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

      5. *-commutative 60.3%

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

      6. fma-neg 60.3%

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

      7. metadata-eval 60.3%

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

   6. Simplified 60.3%

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

      1. associate-/l* 60.1%

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

      2. div-inv 60.3%

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

      3. associate-*l* 60.3%

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

   8. Applied egg-rr 60.3%

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

      1. associate-*r/ 60.0%

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

      2. *-rgt-identity 60.0%

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

      3. associate-/l/ 60.1%

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

      4. associate-*r* 60.2%

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

   10. Simplified 60.2%

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

   11. Taylor expanded in v around 0 56.7%

       \[\leadsto \frac{w}{\color{blue}{\frac{-2.6666666666666665}{w \cdot {r}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 56.7%

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

   13. Simplified 56.7%

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

   if 3.35e136 < r

   1. Initial program 91.1%

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

      1. sub-neg 91.1%

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

      2. +-commutative 91.1%

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

      3. associate--l+ 91.1%

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

      4. associate-/l* 95.5%

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

      5. distribute-neg-frac 95.5%

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

      6. associate-/r/ 95.4%

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

      7. fma-def 95.4%

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

      8. sub-neg 95.4%

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

   3. Simplified 66.4%

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

   4. Taylor expanded in r around inf 64.2%

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

      1. unpow2 64.2%

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

      2. associate-/l* 64.2%

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

      3. *-commutative 64.2%

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

      4. unpow2 64.2%

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

      5. *-commutative 64.2%

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

      6. fma-neg 64.2%

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

      7. metadata-eval 64.2%

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

   6. Simplified 64.2%

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

      1. associate-/l* 64.7%

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

      2. div-inv 64.7%

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

      3. associate-*l* 64.7%

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

   8. Applied egg-rr 64.7%

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

      1. associate-*r/ 64.7%

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

      2. *-rgt-identity 64.7%

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

      3. associate-/l/ 64.7%

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

      4. associate-*r* 64.7%

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

   10. Simplified 64.7%

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

   11. Taylor expanded in v around inf 64.7%

       \[\leadsto \frac{w}{\color{blue}{\frac{-4}{w \cdot {r}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 64.7%

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

   13. Simplified 64.7%

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

   14. Taylor expanded in w around 0 64.7%

       \[\leadsto \frac{w}{\color{blue}{\frac{-4}{w \cdot {r}^{2}}}} \]
   15. Step-by-step derivation

       1. *-commutative 64.7%

          \[\leadsto \frac{w}{\frac{-4}{\color{blue}{{r}^{2} \cdot w}}} \]

       2. associate-/r* 64.7%

          \[\leadsto \frac{w}{\color{blue}{\frac{\frac{-4}{{r}^{2}}}{w}}} \]

       3. unpow2 64.7%

          \[\leadsto \frac{w}{\frac{\frac{-4}{\color{blue}{r \cdot r}}}{w}} \]

       4. associate-/r* 69.4%

          \[\leadsto \frac{w}{\frac{\color{blue}{\frac{\frac{-4}{r}}{r}}}{w}} \]

   16. Simplified 69.4%

       \[\leadsto \frac{w}{\color{blue}{\frac{\frac{\frac{-4}{r}}{r}}{w}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.15 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 6.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{w}{\frac{-2.6666666666666665}{\left(r \cdot r\right) \cdot w}}\\ \mathbf{elif}\;r \leq 3.35 \cdot 10^{+136}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{w}{\frac{\frac{\frac{-4}{r}}{r}}{w}}\\ \end{array} \]

Alternative 14: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;r \leq 3.85 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 6 \cdot 10^{+106}:\\ \;\;\;\;\frac{w \cdot w}{\frac{-2.6666666666666665}{r \cdot r}}\\ \mathbf{elif}\;r \leq 3.35 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{w}{\frac{\frac{\frac{-4}{r}}{r}}{w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5)))
   (if (<= r 3.85e+34)
     t_0
     (if (<= r 6e+106)
       (/ (* w w) (/ -2.6666666666666665 (* r r)))
       (if (<= r 3.35e+136) t_0 (/ w (/ (/ (/ -4.0 r) r) w)))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (r <= 3.85e+34) {
		tmp = t_0;
	} else if (r <= 6e+106) {
		tmp = (w * w) / (-2.6666666666666665 / (r * r));
	} else if (r <= 3.35e+136) {
		tmp = t_0;
	} else {
		tmp = w / (((-4.0 / r) / r) / w);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / (r * r)) + (-1.5d0)
    if (r <= 3.85d+34) then
        tmp = t_0
    else if (r <= 6d+106) then
        tmp = (w * w) / ((-2.6666666666666665d0) / (r * r))
    else if (r <= 3.35d+136) then
        tmp = t_0
    else
        tmp = w / ((((-4.0d0) / r) / r) / w)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (r <= 3.85e+34) {
		tmp = t_0;
	} else if (r <= 6e+106) {
		tmp = (w * w) / (-2.6666666666666665 / (r * r));
	} else if (r <= 3.35e+136) {
		tmp = t_0;
	} else {
		tmp = w / (((-4.0 / r) / r) / w);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if r <= 3.85e+34:
		tmp = t_0
	elif r <= 6e+106:
		tmp = (w * w) / (-2.6666666666666665 / (r * r))
	elif r <= 3.35e+136:
		tmp = t_0
	else:
		tmp = w / (((-4.0 / r) / r) / w)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if (r <= 3.85e+34)
		tmp = t_0;
	elseif (r <= 6e+106)
		tmp = Float64(Float64(w * w) / Float64(-2.6666666666666665 / Float64(r * r)));
	elseif (r <= 3.35e+136)
		tmp = t_0;
	else
		tmp = Float64(w / Float64(Float64(Float64(-4.0 / r) / r) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if (r <= 3.85e+34)
		tmp = t_0;
	elseif (r <= 6e+106)
		tmp = (w * w) / (-2.6666666666666665 / (r * r));
	elseif (r <= 3.35e+136)
		tmp = t_0;
	else
		tmp = w / (((-4.0 / r) / r) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[LessEqual[r, 3.85e+34], t$95$0, If[LessEqual[r, 6e+106], N[(N[(w * w), $MachinePrecision] / N[(-2.6666666666666665 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 3.35e+136], t$95$0, N[(w / N[(N[(N[(-4.0 / r), $MachinePrecision] / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 3.8499999999999999e34 or 6.0000000000000001e106 < r < 3.35e136

   1. Initial program 84.6%

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

      1. sub-neg 84.6%

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

      2. +-commutative 84.6%

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

      3. associate--l+ 84.6%

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

      4. associate-/l* 86.0%

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

      5. distribute-neg-frac 86.0%

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

      6. associate-/r/ 86.0%

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

      7. fma-def 86.0%

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

      8. sub-neg 86.0%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in r around 0 66.8%

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

      1. sub-neg 66.8%

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

      2. associate-*r/ 66.8%

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

      3. metadata-eval 66.8%

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

      4. unpow2 66.8%

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

      5. metadata-eval 66.8%

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

   6. Simplified 66.8%

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

   if 3.8499999999999999e34 < r < 6.0000000000000001e106

   1. Initial program 89.2%

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

      1. sub-neg 89.2%

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

      2. +-commutative 89.2%

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

      3. associate--l+ 89.2%

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

      4. associate-/l* 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. associate-/r/ 100.0%

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

      7. fma-def 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in r around inf 55.0%

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

      1. unpow2 55.0%

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

      2. associate-/l* 60.3%

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

      3. *-commutative 60.3%

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

      4. unpow2 60.3%

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

      5. *-commutative 60.3%

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

      6. fma-neg 60.3%

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

      7. metadata-eval 60.3%

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

   6. Simplified 60.3%

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

   7. Taylor expanded in v around 0 56.8%

      \[\leadsto \frac{w \cdot w}{\color{blue}{\frac{-2.6666666666666665}{{r}^{2}}}} \]
   8. Step-by-step derivation

      1. unpow2 56.8%

         \[\leadsto \frac{w \cdot w}{\frac{-2.6666666666666665}{\color{blue}{r \cdot r}}} \]

   9. Simplified 56.8%

      \[\leadsto \frac{w \cdot w}{\color{blue}{\frac{-2.6666666666666665}{r \cdot r}}} \]

   if 3.35e136 < r

   1. Initial program 91.1%

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

      1. sub-neg 91.1%

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

      2. +-commutative 91.1%

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

      3. associate--l+ 91.1%

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

      4. associate-/l* 95.5%

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

      5. distribute-neg-frac 95.5%

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

      6. associate-/r/ 95.4%

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

      7. fma-def 95.4%

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

      8. sub-neg 95.4%

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

   3. Simplified 66.4%

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

   4. Taylor expanded in r around inf 64.2%

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

      1. unpow2 64.2%

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

      2. associate-/l* 64.2%

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

      3. *-commutative 64.2%

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

      4. unpow2 64.2%

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

      5. *-commutative 64.2%

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

      6. fma-neg 64.2%

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

      7. metadata-eval 64.2%

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

   6. Simplified 64.2%

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

      1. associate-/l* 64.7%

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

      2. div-inv 64.7%

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

      3. associate-*l* 64.7%

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

   8. Applied egg-rr 64.7%

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

      1. associate-*r/ 64.7%

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

      2. *-rgt-identity 64.7%

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

      3. associate-/l/ 64.7%

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

      4. associate-*r* 64.7%

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

   10. Simplified 64.7%

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

   11. Taylor expanded in v around inf 64.7%

       \[\leadsto \frac{w}{\color{blue}{\frac{-4}{w \cdot {r}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 64.7%

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

   13. Simplified 64.7%

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

   14. Taylor expanded in w around 0 64.7%

       \[\leadsto \frac{w}{\color{blue}{\frac{-4}{w \cdot {r}^{2}}}} \]
   15. Step-by-step derivation

       1. *-commutative 64.7%

          \[\leadsto \frac{w}{\frac{-4}{\color{blue}{{r}^{2} \cdot w}}} \]

       2. associate-/r* 64.7%

          \[\leadsto \frac{w}{\color{blue}{\frac{\frac{-4}{{r}^{2}}}{w}}} \]

       3. unpow2 64.7%

          \[\leadsto \frac{w}{\frac{\frac{-4}{\color{blue}{r \cdot r}}}{w}} \]

       4. associate-/r* 69.4%

          \[\leadsto \frac{w}{\frac{\color{blue}{\frac{\frac{-4}{r}}{r}}}{w}} \]

   16. Simplified 69.4%

       \[\leadsto \frac{w}{\color{blue}{\frac{\frac{\frac{-4}{r}}{r}}{w}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.85 \cdot 10^{+34}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 6 \cdot 10^{+106}:\\ \;\;\;\;\frac{w \cdot w}{\frac{-2.6666666666666665}{r \cdot r}}\\ \mathbf{elif}\;r \leq 3.35 \cdot 10^{+136}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{w}{\frac{\frac{\frac{-4}{r}}{r}}{w}}\\ \end{array} \]

Alternative 15: 56.7% 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 86.0%

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

   1. sub-neg 86.0%

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

   2. +-commutative 86.0%

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

   3. associate--l+ 86.0%

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

   4. associate-/l* 88.5%

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

   5. distribute-neg-frac 88.5%

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

   6. associate-/r/ 88.5%

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

   7. fma-def 88.5%

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

   8. sub-neg 88.5%

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

3. Simplified 80.1%

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

4. Taylor expanded in r around 0 56.2%

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

   1. sub-neg 56.2%

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

   2. associate-*r/ 56.2%

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

   3. metadata-eval 56.2%

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

   4. unpow2 56.2%

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

   5. metadata-eval 56.2%

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

6. Simplified 56.2%

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

7. Final simplification 56.2%

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

Alternative 16: 43.2% 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 86.0%

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

   1. associate--l- 86.0%

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

   2. +-commutative 86.0%

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

   3. associate--l+ 86.0%

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

   4. +-commutative 86.0%

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

   5. associate--r+ 86.0%

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

   6. metadata-eval 86.0%

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

   7. associate-*l/ 88.5%

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

   8. *-commutative 88.5%

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

   9. *-commutative 88.5%

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

   10. *-commutative 88.5%

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

3. Simplified 88.5%

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

4. Taylor expanded in r around 0 80.1%

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

   1. unpow2 80.1%

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

   2. *-commutative 80.1%

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

   3. unpow2 80.1%

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

   4. swap-sqr 99.8%

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

   5. unpow2 99.8%

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

   6. *-commutative 99.8%

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

6. Simplified 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in v around 0 92.7%

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

10. Taylor expanded in r around 0 41.2%

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

    1. unpow2 41.2%

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

12. Simplified 41.2%

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

13. Final simplification 41.2%

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

Reproduce

herbie shell --seed 2023230 
(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))