Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.8%

Time: 8.4s

Alternatives: 9

Speedup: 1.7×

• 52.8% 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: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. sub-neg 85.2%

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

   2. +-commutative 85.2%

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

   3. associate--l+ 85.2%

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

   4. associate-/l* 87.8%

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

   5. distribute-neg-frac 87.8%

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

   6. associate-/r/ 87.8%

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

   7. fma-def 87.8%

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

   8. sub-neg 87.8%

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

3. Simplified 79.9%

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

   1. fma-udef 79.9%

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

   2. unswap-sqr 99.8%

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

   3. pow2 99.8%

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

   4. div-inv 99.8%

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

   5. fma-def 99.8%

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

   6. pow2 99.8%

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

   7. pow-flip 99.9%

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

   8. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{-316}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{elif}\;w \cdot w \leq 10^{+297}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 5e-316)
     (+ t_0 (- -1.5 (* 0.375 (* r (* w (* r w))))))
     (if (<= (* w w) 1e+297)
       (+
        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) <= 5e-316) {
		tmp = t_0 + (-1.5 - (0.375 * (r * (w * (r * w)))));
	} else if ((w * w) <= 1e+297) {
		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) <= 5d-316) then
        tmp = t_0 + ((-1.5d0) - (0.375d0 * (r * (w * (r * w)))))
    else if ((w * w) <= 1d+297) 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) <= 5e-316) {
		tmp = t_0 + (-1.5 - (0.375 * (r * (w * (r * w)))));
	} else if ((w * w) <= 1e+297) {
		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) <= 5e-316:
		tmp = t_0 + (-1.5 - (0.375 * (r * (w * (r * w)))))
	elif (w * w) <= 1e+297:
		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) <= 5e-316)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(w * Float64(r * w))))));
	elseif (Float64(w * w) <= 1e+297)
		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) <= 5e-316)
		tmp = t_0 + (-1.5 - (0.375 * (r * (w * (r * w)))));
	elseif ((w * w) <= 1e+297)
		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], 5e-316], 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], 1e+297], 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) < 5.000000017e-316

   1. Initial program 81.7%

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

      1. associate--l- 81.7%

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

      2. +-commutative 81.7%

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

      3. associate--l+ 81.8%

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

      4. +-commutative 81.8%

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

      5. associate--r+ 81.8%

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

      6. metadata-eval 81.8%

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

      7. associate-*l/ 81.8%

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

      8. *-commutative 81.8%

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

      9. *-commutative 81.8%

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

      10. *-commutative 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in r around 0 81.8%

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

      1. *-commutative 81.8%

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

      2. unpow2 81.8%

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

      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 0 63.6%

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

      1. *-commutative 63.6%

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

      2. unpow2 63.6%

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

      3. unpow2 63.6%

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

      4. swap-sqr 92.5%

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

      5. unpow2 92.5%

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

      6. *-commutative 92.5%

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

   9. Simplified 92.5%

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

       1. unpow2 92.5%

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

       2. *-commutative 92.5%

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

       3. *-commutative 92.5%

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

       4. associate-*r* 92.5%

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

       5. *-commutative 92.5%

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

   11. Applied egg-rr 92.5%

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

   if 5.000000017e-316 < (*.f64 w w) < 1e297

   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/ 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   if 1e297 < (*.f64 w w)

   1. Initial program 73.2%

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

      1. associate--l- 73.2%

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

      2. +-commutative 73.2%

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

      3. associate--l+ 73.2%

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

      4. +-commutative 73.2%

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

      5. associate--r+ 73.2%

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

      6. metadata-eval 73.2%

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

      7. associate-*l/ 73.2%

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

      8. *-commutative 73.2%

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

      9. *-commutative 73.2%

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

      10. *-commutative 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in r around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. unpow2 73.2%

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

      3. associate-*r* 95.9%

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

      4. *-commutative 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in v around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. unpow2 73.2%

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

      3. unpow2 73.2%

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

      4. swap-sqr 97.6%

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

      5. unpow2 97.6%

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

      6. *-commutative 97.6%

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

   9. Simplified 97.6%

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

       1. unpow2 97.6%

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

       2. *-commutative 97.6%

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

       3. *-commutative 97.6%

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

   11. Applied egg-rr 97.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{-316}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{elif}\;w \cdot w \leq 10^{+297}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 3: 97.4% accurate, 1.2× speedup

Math

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

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(r * Float64(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[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

   1. associate--l- 85.2%

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

   2. +-commutative 85.2%

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

   3. associate--l+ 85.2%

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

   4. +-commutative 85.2%

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

   5. associate--r+ 85.2%

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

   6. metadata-eval 85.2%

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

   7. associate-*l/ 87.8%

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

   8. *-commutative 87.8%

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

   9. *-commutative 87.8%

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

   10. *-commutative 87.8%

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

3. Simplified 87.8%

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

4. Taylor expanded in r around 0 87.8%

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

   1. *-commutative 87.8%

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

   2. unpow2 87.8%

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

   3. associate-*r* 98.7%

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

   4. *-commutative 98.7%

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

6. Simplified 98.7%

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

7. Final simplification 98.7%

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

Alternative 4: 93.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. associate--l- 85.2%

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

   2. +-commutative 85.2%

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

   3. associate--l+ 85.2%

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

   4. +-commutative 85.2%

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

   5. associate--r+ 85.2%

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

   6. metadata-eval 85.2%

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

   7. associate-*l/ 87.8%

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

   8. *-commutative 87.8%

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

   9. *-commutative 87.8%

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

   10. *-commutative 87.8%

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

3. Simplified 87.8%

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

4. Taylor expanded in r around 0 87.8%

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

   1. *-commutative 87.8%

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

   2. unpow2 87.8%

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

   3. associate-*r* 98.7%

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

   4. *-commutative 98.7%

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

6. Simplified 98.7%

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

7. Taylor expanded in v around 0 77.9%

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

   1. *-commutative 77.9%

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

   2. unpow2 77.9%

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

   3. unpow2 77.9%

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

   4. swap-sqr 93.8%

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

   5. unpow2 93.8%

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

   6. *-commutative 93.8%

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

9. Simplified 93.8%

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

    1. unpow2 93.8%

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

    2. *-commutative 93.8%

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

    3. *-commutative 93.8%

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

11. Applied egg-rr 93.8%

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

12. Final simplification 93.8%

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

Alternative 5: 65.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 9 \cdot 10^{+36}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 9e+36) (+ -1.5 (/ 2.0 (* r r))) (* (* r r) (* -0.25 (* w w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 9e+36) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = (r * r) * (-0.25 * (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) :: tmp
    if (r <= 9d+36) then
        tmp = (-1.5d0) + (2.0d0 / (r * r))
    else
        tmp = (r * r) * ((-0.25d0) * (w * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 9e+36) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = (r * r) * (-0.25 * (w * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 9e+36:
		tmp = -1.5 + (2.0 / (r * r))
	else:
		tmp = (r * r) * (-0.25 * (w * w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 9e+36)
		tmp = Float64(-1.5 + Float64(2.0 / Float64(r * r)));
	else
		tmp = Float64(Float64(r * r) * Float64(-0.25 * Float64(w * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 9e+36)
		tmp = -1.5 + (2.0 / (r * r));
	else
		tmp = (r * r) * (-0.25 * (w * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 9e+36], N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * r), $MachinePrecision] * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8.99999999999999994e36

   1. Initial program 84.5%

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

      1. sub-neg 84.5%

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

      2. +-commutative 84.5%

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

      3. associate--l+ 84.5%

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

      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/ 86.1%

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

      7. fma-def 86.1%

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

      8. sub-neg 86.1%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in r around 0 66.2%

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

      1. sub-neg 66.2%

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

      2. associate-*r/ 66.2%

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

      3. metadata-eval 66.2%

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

      4. unpow2 66.2%

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

      5. metadata-eval 66.2%

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

   6. Simplified 66.2%

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

   if 8.99999999999999994e36 < r

   1. Initial program 87.2%

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

      1. sub-neg 87.2%

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

      2. +-commutative 87.2%

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

      3. associate--l+ 87.2%

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

      4. associate-/l* 92.9%

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

      5. distribute-neg-frac 92.9%

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

      6. associate-/r/ 92.9%

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

      7. fma-def 92.9%

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

      8. sub-neg 92.9%

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

   3. Simplified 74.5%

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

   4. Taylor expanded in v around inf 69.7%

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

      1. associate--l+ 69.7%

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

      2. associate-*r/ 69.7%

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

      3. metadata-eval 69.7%

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

      4. unpow2 69.7%

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

      5. *-commutative 69.7%

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

      6. unpow2 69.7%

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

      7. unpow2 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in r around inf 56.5%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. associate-*r* 56.5%

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

   9. Simplified 56.5%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9 \cdot 10^{+36}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \end{array} \]

Alternative 6: 65.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 8.2 \cdot 10^{+38}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 8.2e+38) (+ -1.5 (/ 2.0 (* r r))) (* -0.375 (* (* r r) (* w w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 8.2e+38) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = -0.375 * ((r * r) * (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) :: tmp
    if (r <= 8.2d+38) then
        tmp = (-1.5d0) + (2.0d0 / (r * r))
    else
        tmp = (-0.375d0) * ((r * r) * (w * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 8.2e+38) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = -0.375 * ((r * r) * (w * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 8.2e+38:
		tmp = -1.5 + (2.0 / (r * r))
	else:
		tmp = -0.375 * ((r * r) * (w * w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 8.2e+38)
		tmp = Float64(-1.5 + Float64(2.0 / Float64(r * r)));
	else
		tmp = Float64(-0.375 * Float64(Float64(r * r) * Float64(w * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 8.2e+38)
		tmp = -1.5 + (2.0 / (r * r));
	else
		tmp = -0.375 * ((r * r) * (w * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 8.2e+38], N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.375 * N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8.2000000000000007e38

   1. Initial program 84.5%

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

      1. sub-neg 84.5%

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

      2. +-commutative 84.5%

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

      3. associate--l+ 84.5%

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

      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/ 86.1%

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

      7. fma-def 86.1%

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

      8. sub-neg 86.1%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in r around 0 66.2%

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

      1. sub-neg 66.2%

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

      2. associate-*r/ 66.2%

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

      3. metadata-eval 66.2%

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

      4. unpow2 66.2%

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

      5. metadata-eval 66.2%

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

   6. Simplified 66.2%

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

   if 8.2000000000000007e38 < r

   1. Initial program 87.2%

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

      1. associate--l- 87.2%

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

      2. +-commutative 87.2%

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

      3. associate--l+ 87.2%

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

      4. +-commutative 87.2%

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

      5. associate--r+ 87.2%

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

      6. metadata-eval 87.2%

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

      7. associate-*l/ 92.9%

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

      8. *-commutative 92.9%

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

      9. *-commutative 92.9%

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

      10. *-commutative 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in r around 0 92.9%

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

      1. *-commutative 92.9%

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

      2. unpow2 92.9%

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

      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 70.8%

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

      1. *-commutative 70.8%

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

      2. unpow2 70.8%

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

      3. unpow2 70.8%

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

      4. swap-sqr 90.2%

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

      5. unpow2 90.2%

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

      6. *-commutative 90.2%

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

   9. Simplified 90.2%

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

       1. unpow2 90.2%

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

       2. *-commutative 90.2%

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

       3. *-commutative 90.2%

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

   11. Applied egg-rr 90.2%

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

   12. Taylor expanded in r around inf 56.9%

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

       1. *-commutative 56.9%

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

       2. unpow2 56.9%

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

       3. unpow2 56.9%

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

       4. *-commutative 56.9%

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

   14. Simplified 56.9%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8.2 \cdot 10^{+38}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \end{array} \]

Alternative 7: 50.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	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 <= 1.15d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.15)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 83.6%

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

      1. sub-neg 83.6%

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

      2. +-commutative 83.6%

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

      3. associate--l+ 83.6%

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

      4. associate-/l* 85.2%

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

      5. distribute-neg-frac 85.2%

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

      6. associate-/r/ 85.2%

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

      7. fma-def 85.2%

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

      8. sub-neg 85.2%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in v around 0 79.2%

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

      1. associate--l+ 79.2%

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

      2. associate-*r/ 79.2%

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

      3. metadata-eval 79.2%

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

      4. unpow2 79.2%

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

      5. *-commutative 79.2%

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

      6. unpow2 79.2%

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

      7. unpow2 79.2%

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

   6. Simplified 79.2%

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

   7. Taylor expanded in r around 0 58.2%

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

      1. unpow2 58.2%

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

   9. Simplified 58.2%

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

   if 1.1499999999999999 < r

   1. Initial program 89.0%

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

      1. sub-neg 89.0%

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

      2. +-commutative 89.0%

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

      3. associate--l+ 89.0%

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

      4. associate-/l* 93.9%

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

      5. distribute-neg-frac 93.9%

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

      6. associate-/r/ 93.9%

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

      7. fma-def 93.9%

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

      8. sub-neg 93.9%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in r around 0 32.8%

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

      1. sub-neg 32.8%

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

      2. associate-*r/ 32.8%

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

      3. metadata-eval 32.8%

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

      4. unpow2 32.8%

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

      5. metadata-eval 32.8%

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

   6. Simplified 32.8%

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

   7. Taylor expanded in r around inf 32.8%

      \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \]

Alternative 8: 57.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. sub-neg 85.2%

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

   2. +-commutative 85.2%

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

   3. associate--l+ 85.2%

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

   4. associate-/l* 87.8%

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

   5. distribute-neg-frac 87.8%

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

   6. associate-/r/ 87.8%

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

   7. fma-def 87.8%

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

   8. sub-neg 87.8%

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

3. Simplified 79.9%

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

4. Taylor expanded in r around 0 56.6%

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

   1. sub-neg 56.6%

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

   2. associate-*r/ 56.6%

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

   3. metadata-eval 56.6%

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

   4. unpow2 56.6%

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

   5. metadata-eval 56.6%

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

6. Simplified 56.6%

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

7. Final simplification 56.6%

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

Alternative 9: 14.0% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -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 = -1.5d0
end function

Java

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

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 85.2%

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

   1. sub-neg 85.2%

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

   2. +-commutative 85.2%

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

   3. associate--l+ 85.2%

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

   4. associate-/l* 87.8%

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

   5. distribute-neg-frac 87.8%

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

   6. associate-/r/ 87.8%

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

   7. fma-def 87.8%

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

   8. sub-neg 87.8%

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

3. Simplified 79.9%

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

4. Taylor expanded in r around 0 56.6%

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

   1. sub-neg 56.6%

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

   2. associate-*r/ 56.6%

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

   3. metadata-eval 56.6%

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

   4. unpow2 56.6%

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

   5. metadata-eval 56.6%

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

6. Simplified 56.6%

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

7. Taylor expanded in r around inf 16.2%

   \[\leadsto \color{blue}{-1.5} \]

8. Final simplification 16.2%

   \[\leadsto -1.5 \]

Reproduce

herbie shell --seed 2023207 
(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))