Rosa's TurbineBenchmark

Percentage Accurate: 84.6% → 99.7%

Time: 13.4s

Alternatives: 15

Speedup: 1.2×

• 52.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.6% 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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (3.0 + (2.0 / (r * r))) - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * ((r * w) / (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 = (3.0d0 + (2.0d0 / (r * r))) - (4.5d0 + ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * ((r * w) / (1.0d0 - v)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

   1. associate--l- 81.5%

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

   2. associate-*l* 75.5%

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

   3. sqr-neg 75.5%

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

   4. associate-*l* 81.5%

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

   5. associate-/l* 84.7%

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

   6. fma-define 84.7%

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

3. Simplified 84.7%

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

   1. add-sqr-sqrt 84.7%

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

   2. *-un-lft-identity 84.7%

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

   3. times-frac 84.7%

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

   4. *-commutative 84.7%

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

   5. sqrt-prod 34.4%

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

   6. *-commutative 34.4%

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

   7. sqrt-prod 34.4%

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

   8. sqrt-prod 16.4%

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

   9. add-sqr-sqrt 25.9%

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

   10. associate-*r* 25.9%

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

   11. add-sqr-sqrt 67.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.375 + v \cdot -0.25\\ t_1 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_2 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2.3 \cdot 10^{+47} \lor \neg \left(v \leq 0.0009\right):\\ \;\;\;\;t\_2 + \left(-1.5 + \frac{t\_0}{\frac{v}{t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(-1.5 - \frac{t\_0}{\frac{1}{t\_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 0.375 (* v -0.25)))
        (t_1 (* (* r w) (* r w)))
        (t_2 (/ 2.0 (* r r))))
   (if (or (<= v -2.3e+47) (not (<= v 0.0009)))
     (+ t_2 (+ -1.5 (/ t_0 (/ v t_1))))
     (+ t_2 (- -1.5 (/ t_0 (/ 1.0 t_1)))))))

C

double code(double v, double w, double r) {
	double t_0 = 0.375 + (v * -0.25);
	double t_1 = (r * w) * (r * w);
	double t_2 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2.3e+47) || !(v <= 0.0009)) {
		tmp = t_2 + (-1.5 + (t_0 / (v / t_1)));
	} else {
		tmp = t_2 + (-1.5 - (t_0 / (1.0 / t_1)));
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 0.375d0 + (v * (-0.25d0))
    t_1 = (r * w) * (r * w)
    t_2 = 2.0d0 / (r * r)
    if ((v <= (-2.3d+47)) .or. (.not. (v <= 0.0009d0))) then
        tmp = t_2 + ((-1.5d0) + (t_0 / (v / t_1)))
    else
        tmp = t_2 + ((-1.5d0) - (t_0 / (1.0d0 / t_1)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 0.375 + (v * -0.25);
	double t_1 = (r * w) * (r * w);
	double t_2 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2.3e+47) || !(v <= 0.0009)) {
		tmp = t_2 + (-1.5 + (t_0 / (v / t_1)));
	} else {
		tmp = t_2 + (-1.5 - (t_0 / (1.0 / t_1)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 0.375 + (v * -0.25)
	t_1 = (r * w) * (r * w)
	t_2 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2.3e+47) or not (v <= 0.0009):
		tmp = t_2 + (-1.5 + (t_0 / (v / t_1)))
	else:
		tmp = t_2 + (-1.5 - (t_0 / (1.0 / t_1)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(0.375 + Float64(v * -0.25))
	t_1 = Float64(Float64(r * w) * Float64(r * w))
	t_2 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2.3e+47) || !(v <= 0.0009))
		tmp = Float64(t_2 + Float64(-1.5 + Float64(t_0 / Float64(v / t_1))));
	else
		tmp = Float64(t_2 + Float64(-1.5 - Float64(t_0 / Float64(1.0 / t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 0.375 + (v * -0.25);
	t_1 = (r * w) * (r * w);
	t_2 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2.3e+47) || ~((v <= 0.0009)))
		tmp = t_2 + (-1.5 + (t_0 / (v / t_1)));
	else
		tmp = t_2 + (-1.5 - (t_0 / (1.0 / t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -2.3e+47], N[Not[LessEqual[v, 0.0009]], $MachinePrecision]], N[(t$95$2 + N[(-1.5 + N[(t$95$0 / N[(v / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(-1.5 - N[(t$95$0 / N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.2999999999999999e47 or 8.9999999999999998e-4 < v

   1. Initial program 75.6%

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

   3. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 81.4%

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

      2. fma-undefine 81.4%

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

      3. *-commutative 81.4%

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

      4. +-commutative 81.4%

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

      5. metadata-eval 81.4%

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

      6. cancel-sign-sub-inv 81.4%

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

      7. associate-*r/ 82.3%

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

      8. *-commutative 82.3%

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

      9. associate-/l* 83.1%

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

      10. associate-*l* 83.1%

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

      11. clear-num 83.1%

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

      12. un-div-inv 83.2%

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

   6. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around inf 99.8%

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

       1. neg-mul-1 99.8%

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

   11. Simplified 99.8%

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

   if -2.2999999999999999e47 < v < 8.9999999999999998e-4

   1. Initial program 86.0%

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

   3. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 86.0%

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

      2. fma-undefine 86.0%

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

      3. *-commutative 86.0%

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

      4. +-commutative 86.0%

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

      5. metadata-eval 86.0%

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

      6. cancel-sign-sub-inv 86.0%

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

      7. associate-*r/ 86.0%

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

      8. *-commutative 86.0%

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

      9. associate-/l* 86.0%

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

      10. associate-*l* 86.0%

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

      11. clear-num 86.0%

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

      12. un-div-inv 86.0%

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

   6. Applied egg-rr 99.3%

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

      1. unpow2 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in v around 0 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.3 \cdot 10^{+47} \lor \neg \left(v \leq 0.0009\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{0.375 + v \cdot -0.25}{\frac{v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{\frac{1}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.75 \cdot 10^{-118}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 240:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.75e-118)
   (+ -1.5 (/ (/ 2.0 r) r))
   (if (<= r 240.0)
     (+
      (/ 2.0 (* r r))
      (- -1.5 (* (* v -0.25) (* r (* (* w w) (/ r (- 1.0 v)))))))
     (-
      3.0
      (+
       4.5
       (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (/ (* r w) (- 1.0 v)))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.75e-118) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else if (r <= 240.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - ((v * -0.25) * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * ((r * w) / (1.0 - v)))));
	}
	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 <= 1.75d-118) then
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    else if (r <= 240.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - ((v * (-0.25d0)) * (r * ((w * w) * (r / (1.0d0 - v))))))
    else
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * ((r * w) / (1.0d0 - v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.75e-118) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else if (r <= 240.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - ((v * -0.25) * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * ((r * w) / (1.0 - v)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.75e-118:
		tmp = -1.5 + ((2.0 / r) / r)
	elif r <= 240.0:
		tmp = (2.0 / (r * r)) + (-1.5 - ((v * -0.25) * (r * ((w * w) * (r / (1.0 - v))))))
	else:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * ((r * w) / (1.0 - v)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.75e-118)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	elseif (r <= 240.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(v * -0.25) * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(1.0 - v)))))));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.75e-118)
		tmp = -1.5 + ((2.0 / r) / r);
	elseif (r <= 240.0)
		tmp = (2.0 / (r * r)) + (-1.5 - ((v * -0.25) * (r * ((w * w) * (r / (1.0 - v))))));
	else
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * ((r * w) / (1.0 - v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.75e-118], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 240.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(v * -0.25), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.75e-118

   1. Initial program 78.2%

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

   3. Simplified 74.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 66.0%

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

      1. +-commutative 66.0%

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

      2. associate--l+ 66.0%

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

      3. div-inv 66.0%

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

      4. pow2 66.0%

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

      5. pow-flip 66.1%

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

      6. metadata-eval 66.1%

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

      7. metadata-eval 66.1%

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

   7. Applied egg-rr 66.1%

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

      1. metadata-eval 66.1%

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

      2. pow-flip 66.0%

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

      3. pow2 66.0%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

      4. div-inv 66.0%

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

      5. associate-/r* 66.1%

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

   9. Applied egg-rr 66.1%

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

   if 1.75e-118 < r < 240

   1. Initial program 84.4%

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

   3. Simplified 92.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 83.7%

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

      1. *-commutative 83.7%

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

   7. Simplified 83.7%

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

   if 240 < r

   1. Initial program 92.7%

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

      1. associate--l- 92.7%

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

      2. associate-*l* 83.1%

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

      3. sqr-neg 83.1%

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

      4. associate-*l* 92.7%

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

      5. associate-/l* 96.3%

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

      6. fma-define 96.4%

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

   3. Simplified 96.3%

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

      1. add-sqr-sqrt 96.2%

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

      2. *-un-lft-identity 96.2%

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

      3. times-frac 96.2%

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

      4. *-commutative 96.2%

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

      5. sqrt-prod 96.2%

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

      6. *-commutative 96.2%

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

      7. sqrt-prod 96.1%

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

      8. sqrt-prod 42.2%

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

      9. add-sqr-sqrt 52.2%

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

      10. associate-*r* 52.3%

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

      11. add-sqr-sqrt 52.3%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in r around inf 99.8%

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

      1. /-rgt-identity 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.75 \cdot 10^{-118}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 240:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.32 \cdot 10^{-118}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 11.8:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.32e-118)
   (+ -1.5 (/ (/ 2.0 r) r))
   (if (<= r 11.8)
     (+
      (/ 2.0 (* r r))
      (- -1.5 (* (* v -0.25) (* r (* (* w w) (/ r (- 1.0 v)))))))
     (-
      (+ 3.0 (* (* r (* w (+ 0.375 (* v -0.25)))) (* w (/ r (+ v -1.0)))))
      4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.32e-118) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else if (r <= 11.8) {
		tmp = (2.0 / (r * r)) + (-1.5 - ((v * -0.25) * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = (3.0 + ((r * (w * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	}
	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 <= 1.32d-118) then
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    else if (r <= 11.8d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - ((v * (-0.25d0)) * (r * ((w * w) * (r / (1.0d0 - v))))))
    else
        tmp = (3.0d0 + ((r * (w * (0.375d0 + (v * (-0.25d0))))) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.32e-118) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else if (r <= 11.8) {
		tmp = (2.0 / (r * r)) + (-1.5 - ((v * -0.25) * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = (3.0 + ((r * (w * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.32e-118:
		tmp = -1.5 + ((2.0 / r) / r)
	elif r <= 11.8:
		tmp = (2.0 / (r * r)) + (-1.5 - ((v * -0.25) * (r * ((w * w) * (r / (1.0 - v))))))
	else:
		tmp = (3.0 + ((r * (w * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.32e-118)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	elseif (r <= 11.8)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(v * -0.25) * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(1.0 - v)))))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(r * Float64(w * Float64(0.375 + Float64(v * -0.25)))) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.32e-118)
		tmp = -1.5 + ((2.0 / r) / r);
	elseif (r <= 11.8)
		tmp = (2.0 / (r * r)) + (-1.5 - ((v * -0.25) * (r * ((w * w) * (r / (1.0 - v))))));
	else
		tmp = (3.0 + ((r * (w * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.32e-118], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 11.8], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(v * -0.25), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(r * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.32000000000000003e-118

   1. Initial program 78.2%

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

   3. Simplified 74.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 66.0%

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

      1. +-commutative 66.0%

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

      2. associate--l+ 66.0%

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

      3. div-inv 66.0%

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

      4. pow2 66.0%

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

      5. pow-flip 66.1%

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

      6. metadata-eval 66.1%

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

      7. metadata-eval 66.1%

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

   7. Applied egg-rr 66.1%

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

      1. metadata-eval 66.1%

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

      2. pow-flip 66.0%

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

      3. pow2 66.0%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

      4. div-inv 66.0%

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

      5. associate-/r* 66.1%

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

   9. Applied egg-rr 66.1%

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

   if 1.32000000000000003e-118 < r < 11.800000000000001

   1. Initial program 84.4%

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

   3. Simplified 92.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 83.7%

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

      1. *-commutative 83.7%

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

   7. Simplified 83.7%

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

   if 11.800000000000001 < r

   1. Initial program 92.7%

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

   3. Taylor expanded in r around inf 92.7%

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

      1. associate-/l* 96.3%

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

      2. cancel-sign-sub-inv 96.3%

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

      3. metadata-eval 96.3%

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

      4. +-commutative 96.3%

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

      5. *-commutative 96.3%

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

      6. fma-undefine 96.3%

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

      7. *-commutative 96.3%

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

      8. *-commutative 96.3%

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

      9. associate-/l* 94.5%

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

      10. *-commutative 94.5%

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

      11. associate-*r/ 94.6%

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

      12. associate-*r* 86.7%

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

      13. associate-*l* 88.5%

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

      14. associate-*r* 88.7%

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

   5. Applied egg-rr 88.7%

      \[\leadsto \left(3 - \color{blue}{\left(\left(\left(0.375 + -0.25 \cdot v\right) \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)}\right) - 4.5 \]

   6. Taylor expanded in r around 0 98.0%

      \[\leadsto \left(3 - \color{blue}{\left(r \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.32 \cdot 10^{-118}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 11.8:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.22 \cdot 10^{-118}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 4.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(r \cdot -0.25\right) \cdot \left(v \cdot w\right)\right) \cdot \left(r \cdot \frac{w}{v}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.22e-118)
   (+ -1.5 (/ (/ 2.0 r) r))
   (if (<= r 4.1e+23)
     (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0)))))))
     (- (+ 3.0 (* (* (* r -0.25) (* v w)) (* r (/ w v)))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.22e-118) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else if (r <= 4.1e+23) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (r * (w / v)))) - 4.5;
	}
	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 <= 1.22d-118) then
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    else if (r <= 4.1d+23) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = (3.0d0 + (((r * (-0.25d0)) * (v * w)) * (r * (w / v)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.22e-118) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else if (r <= 4.1e+23) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (r * (w / v)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.22e-118:
		tmp = -1.5 + ((2.0 / r) / r)
	elif r <= 4.1e+23:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (r * (w / v)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.22e-118)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	elseif (r <= 4.1e+23)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(r * -0.25) * Float64(v * w)) * Float64(r * Float64(w / v)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.22e-118)
		tmp = -1.5 + ((2.0 / r) / r);
	elseif (r <= 4.1e+23)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (r * (w / v)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.22e-118], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 4.1e+23], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(r * -0.25), $MachinePrecision] * N[(v * w), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.2200000000000001e-118

   1. Initial program 78.2%

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

   3. Simplified 74.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 66.0%

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

      1. +-commutative 66.0%

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

      2. associate--l+ 66.0%

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

      3. div-inv 66.0%

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

      4. pow2 66.0%

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

      5. pow-flip 66.1%

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

      6. metadata-eval 66.1%

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

      7. metadata-eval 66.1%

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

   7. Applied egg-rr 66.1%

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

      1. metadata-eval 66.1%

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

      2. pow-flip 66.0%

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

      3. pow2 66.0%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

      4. div-inv 66.0%

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

      5. associate-/r* 66.1%

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

   9. Applied egg-rr 66.1%

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

   if 1.2200000000000001e-118 < r < 4.09999999999999996e23

   1. Initial program 86.7%

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

   3. Simplified 93.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 64.8%

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

   if 4.09999999999999996e23 < r

   1. Initial program 92.4%

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

   3. Taylor expanded in r around inf 92.4%

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

      1. associate-/l* 96.2%

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

      2. cancel-sign-sub-inv 96.2%

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

      3. metadata-eval 96.2%

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

      4. +-commutative 96.2%

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

      5. *-commutative 96.2%

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

      6. fma-undefine 96.2%

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

      7. *-commutative 96.2%

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

      8. *-commutative 96.2%

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

      9. associate-/l* 94.3%

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

      10. *-commutative 94.3%

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

      11. associate-*r/ 94.3%

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

      12. associate-*r* 86.2%

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

      13. associate-*l* 88.0%

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

      14. associate-*r* 88.3%

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

   5. Applied egg-rr 88.3%

      \[\leadsto \left(3 - \color{blue}{\left(\left(\left(0.375 + -0.25 \cdot v\right) \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf 73.1%

      \[\leadsto \left(3 - \color{blue}{\left(-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
   7. Step-by-step derivation

      1. associate-*r* 73.1%

         \[\leadsto \left(3 - \color{blue}{\left(\left(-0.25 \cdot r\right) \cdot \left(v \cdot w\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

      2. *-commutative 73.1%

         \[\leadsto \left(3 - \left(\color{blue}{\left(r \cdot -0.25\right)} \cdot \left(v \cdot w\right)\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

      3. *-commutative 73.1%

         \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \color{blue}{\left(w \cdot v\right)}\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

   8. Simplified 73.1%

      \[\leadsto \left(3 - \color{blue}{\left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

   9. Taylor expanded in v around inf 86.3%

      \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right) \cdot \color{blue}{\left(-1 \cdot \frac{r \cdot w}{v}\right)}\right) - 4.5 \]
   10. Step-by-step derivation

       1. mul-1-neg 86.3%

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

       2. associate-/l* 82.7%

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

       3. distribute-lft-neg-in 82.7%

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

   11. Simplified 82.7%

       \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right) \cdot \color{blue}{\left(\left(-r\right) \cdot \frac{w}{v}\right)}\right) - 4.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.22 \cdot 10^{-118}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 4.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(r \cdot -0.25\right) \cdot \left(v \cdot w\right)\right) \cdot \left(r \cdot \frac{w}{v}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4.5)
   (+ -1.5 (/ (/ 2.0 r) r))
   (-
    (+ 3.0 (* (* r (* w (+ 0.375 (* v -0.25)))) (* w (/ r (+ v -1.0)))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.5) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = (3.0 + ((r * (w * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	}
	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 <= 4.5d0) then
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    else
        tmp = (3.0d0 + ((r * (w * (0.375d0 + (v * (-0.25d0))))) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4.5], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(r * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.5

   1. Initial program 78.6%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. associate--l+ 65.1%

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

      3. div-inv 65.1%

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

      4. pow2 65.1%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

      7. metadata-eval 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. metadata-eval 65.2%

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

      2. pow-flip 65.1%

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

      3. pow2 65.1%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

      4. div-inv 65.1%

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

      5. associate-/r* 65.1%

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

   9. Applied egg-rr 65.1%

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

   if 4.5 < r

   1. Initial program 92.7%

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

   3. Taylor expanded in r around inf 92.7%

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

      1. associate-/l* 96.3%

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

      2. cancel-sign-sub-inv 96.3%

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

      3. metadata-eval 96.3%

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

      4. +-commutative 96.3%

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

      5. *-commutative 96.3%

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

      6. fma-undefine 96.3%

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

      7. *-commutative 96.3%

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

      8. *-commutative 96.3%

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

      9. associate-/l* 94.5%

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

      10. *-commutative 94.5%

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

      11. associate-*r/ 94.6%

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

      12. associate-*r* 86.7%

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

      13. associate-*l* 88.5%

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

      14. associate-*r* 88.7%

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

   5. Applied egg-rr 88.7%

      \[\leadsto \left(3 - \color{blue}{\left(\left(\left(0.375 + -0.25 \cdot v\right) \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)}\right) - 4.5 \]

   6. Taylor expanded in r around 0 98.0%

      \[\leadsto \left(3 - \color{blue}{\left(r \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

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

Alternative 7: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.00025:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{\frac{1 - v}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 0.00025)
   (+ -1.5 (/ (/ 2.0 r) r))
   (-
    3.0
    (+ 4.5 (/ (* (* r w) (+ 0.375 (* v -0.25))) (/ (- 1.0 v) (* r w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00025) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (0.375 + (v * -0.25))) / ((1.0 - v) / (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 <= 0.00025d0) then
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    else
        tmp = 3.0d0 - (4.5d0 + (((r * w) * (0.375d0 + (v * (-0.25d0)))) / ((1.0d0 - v) / (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00025) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (0.375 + (v * -0.25))) / ((1.0 - v) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.00025:
		tmp = -1.5 + ((2.0 / r) / r)
	else:
		tmp = 3.0 - (4.5 + (((r * w) * (0.375 + (v * -0.25))) / ((1.0 - v) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.00025)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * Float64(0.375 + Float64(v * -0.25))) / Float64(Float64(1.0 - v) / Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.00025)
		tmp = -1.5 + ((2.0 / r) / r);
	else
		tmp = 3.0 - (4.5 + (((r * w) * (0.375 + (v * -0.25))) / ((1.0 - v) / (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.00025], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.5000000000000001e-4

   1. Initial program 78.6%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. associate--l+ 65.1%

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

      3. div-inv 65.1%

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

      4. pow2 65.1%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

      7. metadata-eval 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. metadata-eval 65.2%

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

      2. pow-flip 65.1%

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

      3. pow2 65.1%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

      4. div-inv 65.1%

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

      5. associate-/r* 65.1%

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

   9. Applied egg-rr 65.1%

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

   if 2.5000000000000001e-4 < r

   1. Initial program 92.7%

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

      1. associate--l- 92.7%

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

      2. associate-*l* 83.1%

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

      3. sqr-neg 83.1%

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

      4. associate-*l* 92.7%

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

      5. associate-/l* 96.3%

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

      6. fma-define 96.4%

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

   3. Simplified 96.3%

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

      1. add-sqr-sqrt 96.2%

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

      2. *-un-lft-identity 96.2%

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

      3. times-frac 96.2%

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

      4. *-commutative 96.2%

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

      5. sqrt-prod 96.2%

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

      6. *-commutative 96.2%

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

      7. sqrt-prod 96.1%

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

      8. sqrt-prod 42.2%

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

      9. add-sqr-sqrt 52.2%

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

      10. associate-*r* 52.3%

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

      11. add-sqr-sqrt 52.3%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in r around inf 99.8%

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

      1. /-rgt-identity 99.8%

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

      2. associate-*r* 98.0%

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

      3. clear-num 98.1%

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

      4. un-div-inv 98.0%

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

      5. distribute-rgt-in 98.0%

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

      6. metadata-eval 98.0%

         \[\leadsto 3 - \left(\frac{\left(\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      7. *-commutative 98.0%

         \[\leadsto 3 - \left(\frac{\left(0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      8. associate-*l* 98.0%

         \[\leadsto 3 - \left(\frac{\left(0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      9. metadata-eval 98.0%

         \[\leadsto 3 - \left(\frac{\left(0.375 + v \cdot \color{blue}{-0.25}\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

   9. Applied egg-rr 98.0%

      \[\leadsto 3 - \left(\color{blue}{\frac{\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}}} + 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.00025:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{\frac{1 - v}{r \cdot w}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.2× speedup

Math

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

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(0.375 + Float64(v * -0.25)) / Float64(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[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.5%

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

3. Simplified 84.0%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 84.0%

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

   2. fma-undefine 84.0%

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

   3. *-commutative 84.0%

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

   4. +-commutative 84.0%

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

   5. metadata-eval 84.0%

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

   6. cancel-sign-sub-inv 84.0%

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

   7. associate-*r/ 84.4%

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

   8. *-commutative 84.4%

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

   9. associate-/l* 84.7%

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

   10. associate-*l* 84.7%

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

   11. clear-num 84.7%

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

   12. un-div-inv 84.8%

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

6. Applied egg-rr 99.5%

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

   1. unpow2 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 9: 70.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4.5)
   (+ -1.5 (/ (/ 2.0 r) r))
   (- (+ 3.0 (* (* (* r -0.25) (* v w)) (* r (/ w v)))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.5) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (r * (w / v)))) - 4.5;
	}
	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 <= 4.5d0) then
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    else
        tmp = (3.0d0 + (((r * (-0.25d0)) * (v * w)) * (r * (w / v)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.5) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (r * (w / v)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 4.5:
		tmp = -1.5 + ((2.0 / r) / r)
	else:
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (r * (w / v)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 4.5)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(r * -0.25) * Float64(v * w)) * Float64(r * Float64(w / v)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 4.5)
		tmp = -1.5 + ((2.0 / r) / r);
	else
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (r * (w / v)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4.5], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(r * -0.25), $MachinePrecision] * N[(v * w), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.5

   1. Initial program 78.6%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. associate--l+ 65.1%

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

      3. div-inv 65.1%

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

      4. pow2 65.1%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

      7. metadata-eval 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. metadata-eval 65.2%

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

      2. pow-flip 65.1%

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

      3. pow2 65.1%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

      4. div-inv 65.1%

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

      5. associate-/r* 65.1%

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

   9. Applied egg-rr 65.1%

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

   if 4.5 < r

   1. Initial program 92.7%

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

   3. Taylor expanded in r around inf 92.7%

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

      1. associate-/l* 96.3%

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

      2. cancel-sign-sub-inv 96.3%

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

      3. metadata-eval 96.3%

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

      4. +-commutative 96.3%

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

      5. *-commutative 96.3%

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

      6. fma-undefine 96.3%

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

      7. *-commutative 96.3%

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

      8. *-commutative 96.3%

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

      9. associate-/l* 94.5%

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

      10. *-commutative 94.5%

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

      11. associate-*r/ 94.6%

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

      12. associate-*r* 86.7%

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

      13. associate-*l* 88.5%

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

      14. associate-*r* 88.7%

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

   5. Applied egg-rr 88.7%

      \[\leadsto \left(3 - \color{blue}{\left(\left(\left(0.375 + -0.25 \cdot v\right) \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf 72.2%

      \[\leadsto \left(3 - \color{blue}{\left(-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
   7. Step-by-step derivation

      1. associate-*r* 72.2%

         \[\leadsto \left(3 - \color{blue}{\left(\left(-0.25 \cdot r\right) \cdot \left(v \cdot w\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

      2. *-commutative 72.2%

         \[\leadsto \left(3 - \left(\color{blue}{\left(r \cdot -0.25\right)} \cdot \left(v \cdot w\right)\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

      3. *-commutative 72.2%

         \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \color{blue}{\left(w \cdot v\right)}\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

   8. Simplified 72.2%

      \[\leadsto \left(3 - \color{blue}{\left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

   9. Taylor expanded in v around inf 85.3%

      \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right) \cdot \color{blue}{\left(-1 \cdot \frac{r \cdot w}{v}\right)}\right) - 4.5 \]
   10. Step-by-step derivation

       1. mul-1-neg 85.3%

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

       2. associate-/l* 81.8%

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

       3. distribute-lft-neg-in 81.8%

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

   11. Simplified 81.8%

       \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right) \cdot \color{blue}{\left(\left(-r\right) \cdot \frac{w}{v}\right)}\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4.5:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(r \cdot -0.25\right) \cdot \left(v \cdot w\right)\right) \cdot \left(r \cdot \frac{w}{v}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 25.5:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(r \cdot -0.25\right) \cdot \left(v \cdot w\right)\right) \cdot \left(w \cdot \frac{r}{v}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 25.5)
   (+ -1.5 (/ (/ 2.0 r) r))
   (- (+ 3.0 (* (* (* r -0.25) (* v w)) (* w (/ r v)))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 25.5) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (w * (r / v)))) - 4.5;
	}
	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 <= 25.5d0) then
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    else
        tmp = (3.0d0 + (((r * (-0.25d0)) * (v * w)) * (w * (r / v)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 25.5) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (w * (r / v)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 25.5:
		tmp = -1.5 + ((2.0 / r) / r)
	else:
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (w * (r / v)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 25.5)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(r * -0.25) * Float64(v * w)) * Float64(w * Float64(r / v)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 25.5)
		tmp = -1.5 + ((2.0 / r) / r);
	else
		tmp = (3.0 + (((r * -0.25) * (v * w)) * (w * (r / v)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 25.5], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(r * -0.25), $MachinePrecision] * N[(v * w), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 25.5

   1. Initial program 78.6%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. associate--l+ 65.1%

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

      3. div-inv 65.1%

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

      4. pow2 65.1%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

      7. metadata-eval 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. metadata-eval 65.2%

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

      2. pow-flip 65.1%

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

      3. pow2 65.1%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

      4. div-inv 65.1%

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

      5. associate-/r* 65.1%

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

   9. Applied egg-rr 65.1%

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

   if 25.5 < r

   1. Initial program 92.7%

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

   3. Taylor expanded in r around inf 92.7%

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

      1. associate-/l* 96.3%

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

      2. cancel-sign-sub-inv 96.3%

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

      3. metadata-eval 96.3%

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

      4. +-commutative 96.3%

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

      5. *-commutative 96.3%

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

      6. fma-undefine 96.3%

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

      7. *-commutative 96.3%

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

      8. *-commutative 96.3%

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

      9. associate-/l* 94.5%

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

      10. *-commutative 94.5%

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

      11. associate-*r/ 94.6%

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

      12. associate-*r* 86.7%

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

      13. associate-*l* 88.5%

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

      14. associate-*r* 88.7%

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

   5. Applied egg-rr 88.7%

      \[\leadsto \left(3 - \color{blue}{\left(\left(\left(0.375 + -0.25 \cdot v\right) \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf 72.2%

      \[\leadsto \left(3 - \color{blue}{\left(-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
   7. Step-by-step derivation

      1. associate-*r* 72.2%

         \[\leadsto \left(3 - \color{blue}{\left(\left(-0.25 \cdot r\right) \cdot \left(v \cdot w\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

      2. *-commutative 72.2%

         \[\leadsto \left(3 - \left(\color{blue}{\left(r \cdot -0.25\right)} \cdot \left(v \cdot w\right)\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

      3. *-commutative 72.2%

         \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \color{blue}{\left(w \cdot v\right)}\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

   8. Simplified 72.2%

      \[\leadsto \left(3 - \color{blue}{\left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

   9. Taylor expanded in v around inf 83.2%

      \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right) \cdot \left(w \cdot \color{blue}{\left(-1 \cdot \frac{r}{v}\right)}\right)\right) - 4.5 \]
   10. Step-by-step derivation

       1. associate-*r/ 83.2%

          \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right) \cdot \left(w \cdot \color{blue}{\frac{-1 \cdot r}{v}}\right)\right) - 4.5 \]

       2. neg-mul-1 83.2%

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

   11. Simplified 83.2%

       \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right) \cdot \left(w \cdot \color{blue}{\frac{-r}{v}}\right)\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 25.5:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(r \cdot -0.25\right) \cdot \left(v \cdot w\right)\right) \cdot \left(w \cdot \frac{r}{v}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.95 \cdot 10^{-6}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(r \cdot w\right) \cdot 0.375\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.95e-6)
   (+ -1.5 (/ (/ 2.0 r) r))
   (- (+ 3.0 (* (* (* r w) 0.375) (* w (/ r (+ v -1.0))))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.95e-6) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = (3.0 + (((r * w) * 0.375) * (w * (r / (v + -1.0))))) - 4.5;
	}
	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 <= 1.95d-6) then
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    else
        tmp = (3.0d0 + (((r * w) * 0.375d0) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.95e-6) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = (3.0 + (((r * w) * 0.375) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.95e-6:
		tmp = -1.5 + ((2.0 / r) / r)
	else:
		tmp = (3.0 + (((r * w) * 0.375) * (w * (r / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.95e-6)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(r * w) * 0.375) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.95e-6)
		tmp = -1.5 + ((2.0 / r) / r);
	else
		tmp = (3.0 + (((r * w) * 0.375) * (w * (r / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.95e-6], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.95e-6

   1. Initial program 78.6%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. associate--l+ 65.1%

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

      3. div-inv 65.1%

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

      4. pow2 65.1%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

      7. metadata-eval 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. metadata-eval 65.2%

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

      2. pow-flip 65.1%

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

      3. pow2 65.1%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

      4. div-inv 65.1%

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

      5. associate-/r* 65.1%

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

   9. Applied egg-rr 65.1%

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

   if 1.95e-6 < r

   1. Initial program 92.7%

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

   3. Taylor expanded in r around inf 92.7%

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

      1. associate-/l* 96.3%

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

      2. cancel-sign-sub-inv 96.3%

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

      3. metadata-eval 96.3%

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

      4. +-commutative 96.3%

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

      5. *-commutative 96.3%

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

      6. fma-undefine 96.3%

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

      7. *-commutative 96.3%

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

      8. *-commutative 96.3%

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

      9. associate-/l* 94.5%

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

      10. *-commutative 94.5%

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

      11. associate-*r/ 94.6%

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

      12. associate-*r* 86.7%

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

      13. associate-*l* 88.5%

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

      14. associate-*r* 88.7%

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

   5. Applied egg-rr 88.7%

      \[\leadsto \left(3 - \color{blue}{\left(\left(\left(0.375 + -0.25 \cdot v\right) \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)}\right) - 4.5 \]

   6. Taylor expanded in v around 0 74.2%

      \[\leadsto \left(3 - \color{blue}{\left(0.375 \cdot \left(r \cdot w\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.95 \cdot 10^{-6}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(r \cdot w\right) \cdot 0.375\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 122:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{r \cdot \left(w \cdot 0.375\right)}{\frac{v + -1}{r \cdot w}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 122.0)
   (+ -1.5 (/ (/ 2.0 r) r))
   (+ 3.0 (- (/ (* r (* w 0.375)) (/ (+ v -1.0) (* r w))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 122.0) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = 3.0 + (((r * (w * 0.375)) / ((v + -1.0) / (r * w))) - 4.5);
	}
	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 <= 122.0d0) then
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    else
        tmp = 3.0d0 + (((r * (w * 0.375d0)) / ((v + (-1.0d0)) / (r * w))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 122.0) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = 3.0 + (((r * (w * 0.375)) / ((v + -1.0) / (r * w))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 122.0:
		tmp = -1.5 + ((2.0 / r) / r)
	else:
		tmp = 3.0 + (((r * (w * 0.375)) / ((v + -1.0) / (r * w))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 122.0)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(r * Float64(w * 0.375)) / Float64(Float64(v + -1.0) / Float64(r * w))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 122.0)
		tmp = -1.5 + ((2.0 / r) / r);
	else
		tmp = 3.0 + (((r * (w * 0.375)) / ((v + -1.0) / (r * w))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 122.0], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 122

   1. Initial program 78.6%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. associate--l+ 65.1%

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

      3. div-inv 65.1%

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

      4. pow2 65.1%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

      7. metadata-eval 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. metadata-eval 65.2%

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

      2. pow-flip 65.1%

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

      3. pow2 65.1%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

      4. div-inv 65.1%

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

      5. associate-/r* 65.1%

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

   9. Applied egg-rr 65.1%

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

   if 122 < r

   1. Initial program 92.7%

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

      1. associate--l- 92.7%

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

      2. associate-*l* 83.1%

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

      3. sqr-neg 83.1%

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

      4. associate-*l* 92.7%

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

      5. associate-/l* 96.3%

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

      6. fma-define 96.4%

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

   3. Simplified 96.3%

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

      1. add-sqr-sqrt 96.2%

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

      2. *-un-lft-identity 96.2%

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

      3. times-frac 96.2%

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

      4. *-commutative 96.2%

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

      5. sqrt-prod 96.2%

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

      6. *-commutative 96.2%

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

      7. sqrt-prod 96.1%

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

      8. sqrt-prod 42.2%

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

      9. add-sqr-sqrt 52.2%

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

      10. associate-*r* 52.3%

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

      11. add-sqr-sqrt 52.3%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in r around inf 99.8%

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

      1. /-rgt-identity 99.8%

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

      2. associate-*r* 98.0%

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

      3. clear-num 98.1%

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

      4. un-div-inv 98.0%

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

      5. distribute-rgt-in 98.0%

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

      6. metadata-eval 98.0%

         \[\leadsto 3 - \left(\frac{\left(\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      7. *-commutative 98.0%

         \[\leadsto 3 - \left(\frac{\left(0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      8. associate-*l* 98.0%

         \[\leadsto 3 - \left(\frac{\left(0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      9. metadata-eval 98.0%

         \[\leadsto 3 - \left(\frac{\left(0.375 + v \cdot \color{blue}{-0.25}\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

   9. Applied egg-rr 98.0%

      \[\leadsto 3 - \left(\color{blue}{\frac{\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}}} + 4.5\right) \]

   10. Taylor expanded in v around 0 74.2%

       \[\leadsto 3 - \left(\frac{\color{blue}{0.375 \cdot \left(r \cdot w\right)}}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]
   11. Step-by-step derivation

       1. *-commutative 74.2%

          \[\leadsto 3 - \left(\frac{\color{blue}{\left(r \cdot w\right) \cdot 0.375}}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

       2. associate-*l* 74.2%

          \[\leadsto 3 - \left(\frac{\color{blue}{r \cdot \left(w \cdot 0.375\right)}}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

   12. Simplified 74.2%

       \[\leadsto 3 - \left(\frac{\color{blue}{r \cdot \left(w \cdot 0.375\right)}}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 122:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{r \cdot \left(w \cdot 0.375\right)}{\frac{v + -1}{r \cdot w}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 240:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(\left(r \cdot -0.25\right) \cdot \left(v \cdot w\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 240.0)
   (+ -1.5 (/ (/ 2.0 r) r))
   (- (- 3.0 (* (* r w) (* (* r -0.25) (* v w)))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 240.0) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = (3.0 - ((r * w) * ((r * -0.25) * (v * w)))) - 4.5;
	}
	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 <= 240.0d0) then
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    else
        tmp = (3.0d0 - ((r * w) * ((r * (-0.25d0)) * (v * w)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 240.0) {
		tmp = -1.5 + ((2.0 / r) / r);
	} else {
		tmp = (3.0 - ((r * w) * ((r * -0.25) * (v * w)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 240.0:
		tmp = -1.5 + ((2.0 / r) / r)
	else:
		tmp = (3.0 - ((r * w) * ((r * -0.25) * (v * w)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 240.0)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r * w) * Float64(Float64(r * -0.25) * Float64(v * w)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 240.0)
		tmp = -1.5 + ((2.0 / r) / r);
	else
		tmp = (3.0 - ((r * w) * ((r * -0.25) * (v * w)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 240.0], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(r * w), $MachinePrecision] * N[(N[(r * -0.25), $MachinePrecision] * N[(v * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 240

   1. Initial program 78.6%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. associate--l+ 65.1%

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

      3. div-inv 65.1%

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

      4. pow2 65.1%

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

      5. pow-flip 65.2%

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

      6. metadata-eval 65.2%

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

      7. metadata-eval 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. metadata-eval 65.2%

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

      2. pow-flip 65.1%

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

      3. pow2 65.1%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

      4. div-inv 65.1%

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

      5. associate-/r* 65.1%

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

   9. Applied egg-rr 65.1%

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

   if 240 < r

   1. Initial program 92.7%

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

   3. Taylor expanded in r around inf 92.7%

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

      1. associate-/l* 96.3%

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

      2. cancel-sign-sub-inv 96.3%

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

      3. metadata-eval 96.3%

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

      4. +-commutative 96.3%

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

      5. *-commutative 96.3%

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

      6. fma-undefine 96.3%

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

      7. *-commutative 96.3%

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

      8. *-commutative 96.3%

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

      9. associate-/l* 94.5%

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

      10. *-commutative 94.5%

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

      11. associate-*r/ 94.6%

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

      12. associate-*r* 86.7%

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

      13. associate-*l* 88.5%

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

      14. associate-*r* 88.7%

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

   5. Applied egg-rr 88.7%

      \[\leadsto \left(3 - \color{blue}{\left(\left(\left(0.375 + -0.25 \cdot v\right) \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf 72.2%

      \[\leadsto \left(3 - \color{blue}{\left(-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
   7. Step-by-step derivation

      1. associate-*r* 72.2%

         \[\leadsto \left(3 - \color{blue}{\left(\left(-0.25 \cdot r\right) \cdot \left(v \cdot w\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

      2. *-commutative 72.2%

         \[\leadsto \left(3 - \left(\color{blue}{\left(r \cdot -0.25\right)} \cdot \left(v \cdot w\right)\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

      3. *-commutative 72.2%

         \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \color{blue}{\left(w \cdot v\right)}\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

   8. Simplified 72.2%

      \[\leadsto \left(3 - \color{blue}{\left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]

   9. Taylor expanded in v around 0 45.7%

      \[\leadsto \left(3 - \left(\left(r \cdot -0.25\right) \cdot \left(w \cdot v\right)\right) \cdot \left(w \cdot \color{blue}{r}\right)\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 240:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(\left(r \cdot -0.25\right) \cdot \left(v \cdot w\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.6% accurate, 4.1× speedup

Math

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

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(Float64(2.0 / 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[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.5%

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

3. Simplified 76.9%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 56.4%

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

   1. +-commutative 56.4%

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

   2. associate--l+ 56.4%

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

   3. div-inv 56.4%

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

   4. pow2 56.4%

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

   5. pow-flip 56.5%

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

   6. metadata-eval 56.5%

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

   7. metadata-eval 56.5%

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

7. Applied egg-rr 56.5%

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

   1. metadata-eval 56.5%

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

   2. pow-flip 56.4%

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

   3. pow2 56.4%

      \[\leadsto 2 \cdot \frac{1}{\color{blue}{r \cdot r}} + -1.5 \]

   4. div-inv 56.4%

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

   5. associate-/r* 56.4%

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

9. Applied egg-rr 56.4%

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

10. Final simplification 56.4%

    \[\leadsto -1.5 + \frac{\frac{2}{r}}{r} \]
11. Add Preprocessing

Alternative 15: 13.9% 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 81.5%

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

3. Simplified 76.9%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 56.4%

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

6. Taylor expanded in r around inf 13.8%

   \[\leadsto \color{blue}{-1.5} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024119 
(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))