Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 97.7%

Time: 17.9s

Alternatives: 7

Speedup: 1.5×

• 53.6% 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: 84.9% 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: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := v \cdot -0.25 + 0.375\\ \mathbf{if}\;v \leq -2 \cdot 10^{+89}:\\ \;\;\;\;\left(t\_0 + 3\right) + \left(\left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right) - 4.5\right)\\ \mathbf{elif}\;v \leq 6.8 \cdot 10^{-22}:\\ \;\;\;\;t\_0 + \left(-1.5 + \frac{t\_1}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + \frac{t\_1}{\frac{\frac{v}{r \cdot w}}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ (* v -0.25) 0.375)))
   (if (<= v -2e+89)
     (+ (+ t_0 3.0) (- (* (* v -0.25) (* (* r w) (/ (* r w) v))) 4.5))
     (if (<= v 6.8e-22)
       (+ t_0 (+ -1.5 (/ t_1 (/ (/ (/ -1.0 r) w) (* r w)))))
       (+ t_0 (+ -1.5 (/ t_1 (/ (/ v (* r w)) (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (v * -0.25) + 0.375;
	double tmp;
	if (v <= -2e+89) {
		tmp = (t_0 + 3.0) + (((v * -0.25) * ((r * w) * ((r * w) / v))) - 4.5);
	} else if (v <= 6.8e-22) {
		tmp = t_0 + (-1.5 + (t_1 / (((-1.0 / r) / w) / (r * w))));
	} else {
		tmp = t_0 + (-1.5 + (t_1 / ((v / (r * w)) / (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (v * (-0.25d0)) + 0.375d0
    if (v <= (-2d+89)) then
        tmp = (t_0 + 3.0d0) + (((v * (-0.25d0)) * ((r * w) * ((r * w) / v))) - 4.5d0)
    else if (v <= 6.8d-22) then
        tmp = t_0 + ((-1.5d0) + (t_1 / ((((-1.0d0) / r) / w) / (r * w))))
    else
        tmp = t_0 + ((-1.5d0) + (t_1 / ((v / (r * w)) / (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (v * -0.25) + 0.375;
	double tmp;
	if (v <= -2e+89) {
		tmp = (t_0 + 3.0) + (((v * -0.25) * ((r * w) * ((r * w) / v))) - 4.5);
	} else if (v <= 6.8e-22) {
		tmp = t_0 + (-1.5 + (t_1 / (((-1.0 / r) / w) / (r * w))));
	} else {
		tmp = t_0 + (-1.5 + (t_1 / ((v / (r * w)) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (v * -0.25) + 0.375
	tmp = 0
	if v <= -2e+89:
		tmp = (t_0 + 3.0) + (((v * -0.25) * ((r * w) * ((r * w) / v))) - 4.5)
	elif v <= 6.8e-22:
		tmp = t_0 + (-1.5 + (t_1 / (((-1.0 / r) / w) / (r * w))))
	else:
		tmp = t_0 + (-1.5 + (t_1 / ((v / (r * w)) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(v * -0.25) + 0.375)
	tmp = 0.0
	if (v <= -2e+89)
		tmp = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(v * -0.25) * Float64(Float64(r * w) * Float64(Float64(r * w) / v))) - 4.5));
	elseif (v <= 6.8e-22)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(t_1 / Float64(Float64(Float64(-1.0 / r) / w) / Float64(r * w)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(t_1 / Float64(Float64(v / Float64(r * w)) / Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (v * -0.25) + 0.375;
	tmp = 0.0;
	if (v <= -2e+89)
		tmp = (t_0 + 3.0) + (((v * -0.25) * ((r * w) * ((r * w) / v))) - 4.5);
	elseif (v <= 6.8e-22)
		tmp = t_0 + (-1.5 + (t_1 / (((-1.0 / r) / w) / (r * w))));
	else
		tmp = t_0 + (-1.5 + (t_1 / ((v / (r * w)) / (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]}, If[LessEqual[v, -2e+89], N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 6.8e-22], N[(t$95$0 + N[(-1.5 + N[(t$95$1 / N[(N[(N[(-1.0 / r), $MachinePrecision] / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(t$95$1 / N[(N[(v / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.99999999999999999e89

   1. Initial program 78.0%

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

   3. Simplified 86.8%

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

      1. associate-/l* 86.8%

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

      2. *-commutative 86.8%

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

      3. associate-*r/ 86.6%

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

      4. associate-*l* 96.1%

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

      5. associate-*r* 96.1%

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

      6. *-commutative 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in v around inf 98.0%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
   8. Step-by-step derivation

      1. *-commutative 98.0%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(v \cdot -0.25\right)} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]

   9. Simplified 98.0%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(v \cdot -0.25\right)} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]

   10. Taylor expanded in v around inf 99.8%

       \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\left(v \cdot -0.25\right) \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\left(-1 \cdot \frac{r \cdot w}{v}\right)}\right) + 4.5\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 99.8%

          \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\left(v \cdot -0.25\right) \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\frac{-1 \cdot \left(r \cdot w\right)}{v}}\right) + 4.5\right) \]

       2. mul-1-neg 99.8%

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

       3. *-commutative 99.8%

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

       4. distribute-rgt-neg-in 99.8%

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

   12. Simplified 99.8%

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

   if -1.99999999999999999e89 < v < 6.7999999999999997e-22

   1. Initial program 91.1%

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

   3. Simplified 91.1%

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

      1. fma-undefine 91.1%

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

      2. *-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. metadata-eval 91.1%

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

      5. cancel-sign-sub-inv 91.1%

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

      6. associate-*r/ 91.1%

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

      7. *-commutative 91.1%

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

      8. associate-/l* 91.1%

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

      9. clear-num 91.1%

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

      10. un-div-inv 91.1%

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

      11. cancel-sign-sub-inv 91.1%

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

      12. metadata-eval 91.1%

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

      13. +-commutative 91.1%

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

      14. distribute-rgt-in 91.1%

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

      15. *-commutative 91.1%

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

      16. associate-*l* 91.1%

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

      17. metadata-eval 91.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      18. metadata-eval 91.1%

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

   6. Applied egg-rr 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in v around 0 99.8%

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

       1. associate-/r* 99.8%

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

   13. Simplified 99.8%

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

   if 6.7999999999999997e-22 < v

   1. Initial program 85.9%

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

   3. Simplified 89.9%

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

      1. fma-undefine 89.9%

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

      2. *-commutative 89.9%

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

      3. +-commutative 89.9%

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

      4. metadata-eval 89.9%

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

      5. cancel-sign-sub-inv 89.9%

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

      6. associate-*r/ 89.9%

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

      7. *-commutative 89.9%

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

      8. associate-/l* 89.9%

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

      9. clear-num 89.9%

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

      10. un-div-inv 89.9%

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

      11. cancel-sign-sub-inv 89.9%

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

      12. metadata-eval 89.9%

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

      13. +-commutative 89.9%

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

      14. distribute-rgt-in 89.9%

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

      15. *-commutative 89.9%

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

      16. associate-*l* 89.9%

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

      17. metadata-eval 89.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      18. metadata-eval 89.9%

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

   6. Applied egg-rr 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

      2. *-lft-identity 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in v around inf 99.9%

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

       1. mul-1-neg 99.9%

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

       2. distribute-neg-frac2 99.9%

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

       3. *-commutative 99.9%

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

       4. distribute-rgt-neg-in 99.9%

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

   13. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+89}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) + \left(\left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right) - 4.5\right)\\ \mathbf{elif}\;v \leq 6.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{v}{r \cdot w}}{r \cdot w}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := v \cdot -0.25 + 0.375\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6.5 \cdot 10^{+87} \lor \neg \left(v \leq 6.8 \cdot 10^{-22}\right):\\ \;\;\;\;t\_1 + \left(-1.5 + \frac{t\_0}{\frac{\frac{v}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 + \frac{t\_0}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (* v -0.25) 0.375)) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -6.5e+87) (not (<= v 6.8e-22)))
     (+ t_1 (+ -1.5 (/ t_0 (/ (/ v (* r w)) (* r w)))))
     (+ t_1 (+ -1.5 (/ t_0 (/ (/ (/ -1.0 r) w) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = (v * -0.25) + 0.375;
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -6.5e+87) || !(v <= 6.8e-22)) {
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))));
	} else {
		tmp = t_1 + (-1.5 + (t_0 / (((-1.0 / r) / w) / (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (v * (-0.25d0)) + 0.375d0
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-6.5d+87)) .or. (.not. (v <= 6.8d-22))) then
        tmp = t_1 + ((-1.5d0) + (t_0 / ((v / (r * w)) / (r * w))))
    else
        tmp = t_1 + ((-1.5d0) + (t_0 / ((((-1.0d0) / r) / w) / (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (v * -0.25) + 0.375;
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -6.5e+87) || !(v <= 6.8e-22)) {
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))));
	} else {
		tmp = t_1 + (-1.5 + (t_0 / (((-1.0 / r) / w) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (v * -0.25) + 0.375
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -6.5e+87) or not (v <= 6.8e-22):
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))))
	else:
		tmp = t_1 + (-1.5 + (t_0 / (((-1.0 / r) / w) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(v * -0.25) + 0.375)
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -6.5e+87) || !(v <= 6.8e-22))
		tmp = Float64(t_1 + Float64(-1.5 + Float64(t_0 / Float64(Float64(v / Float64(r * w)) / Float64(r * w)))));
	else
		tmp = Float64(t_1 + Float64(-1.5 + Float64(t_0 / Float64(Float64(Float64(-1.0 / r) / w) / Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (v * -0.25) + 0.375;
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -6.5e+87) || ~((v <= 6.8e-22)))
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))));
	else
		tmp = t_1 + (-1.5 + (t_0 / (((-1.0 / r) / w) / (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -6.5e+87], N[Not[LessEqual[v, 6.8e-22]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 + N[(t$95$0 / N[(N[(v / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 + N[(t$95$0 / N[(N[(N[(-1.0 / r), $MachinePrecision] / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -6.5000000000000002e87 or 6.7999999999999997e-22 < v

   1. Initial program 82.5%

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

   3. Simplified 88.5%

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

      1. fma-undefine 88.5%

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

      2. *-commutative 88.5%

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

      3. +-commutative 88.5%

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

      4. metadata-eval 88.5%

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

      5. cancel-sign-sub-inv 88.5%

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

      6. associate-*r/ 88.6%

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

      7. *-commutative 88.6%

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

      8. associate-/l* 88.6%

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

      9. clear-num 88.6%

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

      10. un-div-inv 88.6%

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

      11. cancel-sign-sub-inv 88.6%

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

      12. metadata-eval 88.6%

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

      13. +-commutative 88.6%

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

      14. distribute-rgt-in 88.6%

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

      15. *-commutative 88.6%

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

      16. associate-*l* 89.4%

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

      17. metadata-eval 89.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      18. metadata-eval 89.4%

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

   6. Applied egg-rr 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in v around inf 99.8%

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

       1. mul-1-neg 99.8%

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

       2. distribute-neg-frac2 99.8%

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

       3. *-commutative 99.8%

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

       4. distribute-rgt-neg-in 99.8%

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

   13. Simplified 99.8%

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

   if -6.5000000000000002e87 < v < 6.7999999999999997e-22

   1. Initial program 91.1%

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

   3. Simplified 91.1%

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

      1. fma-undefine 91.1%

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

      2. *-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. metadata-eval 91.1%

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

      5. cancel-sign-sub-inv 91.1%

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

      6. associate-*r/ 91.1%

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

      7. *-commutative 91.1%

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

      8. associate-/l* 91.1%

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

      9. clear-num 91.1%

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

      10. un-div-inv 91.1%

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

      11. cancel-sign-sub-inv 91.1%

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

      12. metadata-eval 91.1%

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

      13. +-commutative 91.1%

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

      14. distribute-rgt-in 91.1%

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

      15. *-commutative 91.1%

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

      16. associate-*l* 91.1%

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

      17. metadata-eval 91.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      18. metadata-eval 91.1%

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

   6. Applied egg-rr 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in v around 0 99.8%

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

       1. associate-/r* 99.8%

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

   13. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6.5 \cdot 10^{+87} \lor \neg \left(v \leq 6.8 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{v}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Simplified 89.9%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. unpow2 99.8%

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

   3. times-frac 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. *-lft-identity 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 - (((v * -0.25) + 0.375) / ((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) - (((v * (-0.25d0)) + 0.375d0) / ((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 - (((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w)))));
}

Python

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

Julia

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

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $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 87.0%

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

3. Simplified 89.9%

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

   1. fma-undefine 89.9%

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

   2. *-commutative 89.9%

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

   3. +-commutative 89.9%

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

   4. metadata-eval 89.9%

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

   5. cancel-sign-sub-inv 89.9%

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

   6. associate-*r/ 89.9%

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

   7. *-commutative 89.9%

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

   8. associate-/l* 89.9%

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

   9. clear-num 89.9%

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

   10. un-div-inv 89.9%

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

   11. cancel-sign-sub-inv 89.9%

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

   12. metadata-eval 89.9%

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

   13. +-commutative 89.9%

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

   14. distribute-rgt-in 89.9%

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

   15. *-commutative 89.9%

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

   16. associate-*l* 90.3%

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

   17. metadata-eval 90.3%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   18. metadata-eval 90.3%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 93.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\frac{2}{r \cdot r} + 3\right) - \left(4.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (- (+ (/ 2.0 (* r r)) 3.0) (+ 4.5 (* 0.375 (* (* r w) (* r w))))))

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + 3.0) - (4.5 + (0.375 * ((r * w) * (r * w))));
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / (r * r)) + 3.0d0) - (4.5d0 + (0.375d0 * ((r * w) * (r * w))))
end function

Java

public static double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + 3.0) - (4.5 + (0.375 * ((r * w) * (r * w))));
}

Python

def code(v, w, r):
	return ((2.0 / (r * r)) + 3.0) - (4.5 + (0.375 * ((r * w) * (r * w))))

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(4.5 + Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((2.0 / (r * r)) + 3.0) - (4.5 + (0.375 * ((r * w) * (r * w))));
end

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Simplified 89.9%

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

   1. associate-/l* 89.9%

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

   2. *-commutative 89.9%

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

   3. associate-*r/ 89.9%

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

   4. associate-*l* 98.3%

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

   5. associate-*r* 99.0%

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

   6. *-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in v around 0 82.7%

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

8. Taylor expanded in v around 0 95.1%

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

9. Final simplification 95.1%

   \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(4.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \]
10. Add Preprocessing

Alternative 6: 93.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{2}{r \cdot r} + 3\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (- (- (+ (/ 2.0 (* r r)) 3.0) (* (* r w) (* r (* w 0.375)))) 4.5))

C

double code(double v, double w, double r) {
	return (((2.0 / (r * r)) + 3.0) - ((r * w) * (r * (w * 0.375)))) - 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 = (((2.0d0 / (r * r)) + 3.0d0) - ((r * w) * (r * (w * 0.375d0)))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return (((2.0 / (r * r)) + 3.0) - ((r * w) * (r * (w * 0.375)))) - 4.5;
}

Python

def code(v, w, r):
	return (((2.0 / (r * r)) + 3.0) - ((r * w) * (r * (w * 0.375)))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(r * w) * Float64(r * Float64(w * 0.375)))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = (((2.0 / (r * r)) + 3.0) - ((r * w) * (r * (w * 0.375)))) - 4.5;
end

Wolfram

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

Derivation

1. Initial program 87.0%

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

4. Applied egg-rr 94.9%

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

5. Taylor expanded in v around 0 82.6%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.375 \cdot \left(r \cdot w\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
6. Step-by-step derivation

   1. associate-*r* 82.6%

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

   2. *-commutative 82.6%

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

   3. associate-*l* 82.7%

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

7. Simplified 82.7%

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

8. Taylor expanded in v around 0 95.1%

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

9. Final simplification 95.1%

   \[\leadsto \left(\left(\frac{2}{r \cdot r} + 3\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right) - 4.5 \]
10. Add Preprocessing

Alternative 7: 93.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (- (- (+ 3.0 (/ (/ 2.0 r) r)) (* (* r w) (* r (* w 0.375)))) 4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + ((2.0 / r) / r)) - ((r * w) * (r * (w * 0.375)))) - 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)) - ((r * w) * (r * (w * 0.375d0)))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + ((2.0 / r) / r)) - ((r * w) * (r * (w * 0.375)))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + ((2.0 / r) / r)) - ((r * w) * (r * (w * 0.375)))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - Float64(Float64(r * w) * Float64(r * Float64(w * 0.375)))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + ((2.0 / r) / r)) - ((r * w) * (r * (w * 0.375)))) - 4.5;
end

Wolfram

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

Derivation

1. Initial program 87.0%

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

4. Applied egg-rr 94.9%

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

5. Taylor expanded in v around 0 82.6%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.375 \cdot \left(r \cdot w\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
6. Step-by-step derivation

   1. associate-*r* 82.6%

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

   2. *-commutative 82.6%

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

   3. associate-*l* 82.7%

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

7. Simplified 82.7%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(r \cdot \left(0.375 \cdot w\right)\right)} \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
8. Step-by-step derivation

   1. associate-/r* 82.7%

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

   2. div-inv 82.6%

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

9. Applied egg-rr 82.6%

   \[\leadsto \left(\left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - \left(r \cdot \left(0.375 \cdot w\right)\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
10. Step-by-step derivation

    1. associate-*r/ 82.7%

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

    2. *-rgt-identity 82.7%

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

11. Simplified 82.7%

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

12. Taylor expanded in v around 0 95.1%

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

13. Final simplification 95.1%

    \[\leadsto \left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right) - 4.5 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(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))