Rosa's TurbineBenchmark

Percentage Accurate: 83.8% → 98.6%

Time: 19.3s

Alternatives: 9

Speedup: 1.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\\ \mathbf{if}\;v \leq -2.7 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{r \cdot \frac{r}{2}} + t\_0\\ \mathbf{elif}\;v \leq 9 \cdot 10^{-23}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left({\left(r \cdot w\right)}^{2} \cdot \left(0.375 + v \cdot 0.125\right) + 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- -1.5 (* (* (* r w) (* r w)) 0.25))))
   (if (<= v -2.7e+22)
     (+ (/ 1.0 (* r (/ r 2.0))) t_0)
     (if (<= v 9e-23)
       (-
        (+ 3.0 (/ 2.0 (* r r)))
        (+ (* (pow (* r w) 2.0) (+ 0.375 (* v 0.125))) 4.5))
       (+ (/ (/ 2.0 r) r) t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = -1.5 - (((r * w) * (r * w)) * 0.25);
	double tmp;
	if (v <= -2.7e+22) {
		tmp = (1.0 / (r * (r / 2.0))) + t_0;
	} else if (v <= 9e-23) {
		tmp = (3.0 + (2.0 / (r * r))) - ((pow((r * w), 2.0) * (0.375 + (v * 0.125))) + 4.5);
	} else {
		tmp = ((2.0 / r) / r) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.5d0) - (((r * w) * (r * w)) * 0.25d0)
    if (v <= (-2.7d+22)) then
        tmp = (1.0d0 / (r * (r / 2.0d0))) + t_0
    else if (v <= 9d-23) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - ((((r * w) ** 2.0d0) * (0.375d0 + (v * 0.125d0))) + 4.5d0)
    else
        tmp = ((2.0d0 / r) / r) + t_0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = -1.5 - (((r * w) * (r * w)) * 0.25);
	double tmp;
	if (v <= -2.7e+22) {
		tmp = (1.0 / (r * (r / 2.0))) + t_0;
	} else if (v <= 9e-23) {
		tmp = (3.0 + (2.0 / (r * r))) - ((Math.pow((r * w), 2.0) * (0.375 + (v * 0.125))) + 4.5);
	} else {
		tmp = ((2.0 / r) / r) + t_0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = -1.5 - (((r * w) * (r * w)) * 0.25)
	tmp = 0
	if v <= -2.7e+22:
		tmp = (1.0 / (r * (r / 2.0))) + t_0
	elif v <= 9e-23:
		tmp = (3.0 + (2.0 / (r * r))) - ((math.pow((r * w), 2.0) * (0.375 + (v * 0.125))) + 4.5)
	else:
		tmp = ((2.0 / r) / r) + t_0
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25))
	tmp = 0.0
	if (v <= -2.7e+22)
		tmp = Float64(Float64(1.0 / Float64(r * Float64(r / 2.0))) + t_0);
	elseif (v <= 9e-23)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64((Float64(r * w) ^ 2.0) * Float64(0.375 + Float64(v * 0.125))) + 4.5));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = -1.5 - (((r * w) * (r * w)) * 0.25);
	tmp = 0.0;
	if (v <= -2.7e+22)
		tmp = (1.0 / (r * (r / 2.0))) + t_0;
	elseif (v <= 9e-23)
		tmp = (3.0 + (2.0 / (r * r))) - ((((r * w) ^ 2.0) * (0.375 + (v * 0.125))) + 4.5);
	else
		tmp = ((2.0 / r) / r) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2.7e+22], N[(N[(1.0 / N[(r * N[(r / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[v, 9e-23], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.375 + N[(v * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.7000000000000002e22

   1. Initial program 86.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in v around inf 92.1%

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

      1. *-commutative 92.1%

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

      2. *-commutative 92.1%

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

      3. unpow2 92.1%

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

      4. unpow2 92.1%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. clear-num 99.8%

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

       2. inv-pow 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. unpow-1 99.8%

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

       2. associate-/r/ 99.8%

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

   13. Simplified 99.8%

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

   if -2.7000000000000002e22 < v < 8.9999999999999995e-23

   1. Initial program 87.1%

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

   3. Simplified 87.0%

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

      1. metadata-eval 87.0%

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

      2. cancel-sign-sub-inv 87.0%

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

      3. associate-*r* 96.8%

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

      4. *-commutative 96.8%

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

      5. *-commutative 96.8%

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

      6. *-commutative 96.8%

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

      7. associate-*l* 87.0%

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

      8. associate-/l* 87.1%

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

      9. clear-num 87.0%

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

      10. inv-pow 87.0%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{1}{8 \cdot \frac{1 - v}{\mathsf{fma}\left(v, -2, 3\right) \cdot {\left(w \cdot r\right)}^{2}}}} + 4.5\right) \]

      2. associate-*r/ 99.8%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{\color{blue}{\frac{8 \cdot \left(1 - v\right)}{\mathsf{fma}\left(v, -2, 3\right) \cdot {\left(w \cdot r\right)}^{2}}}} + 4.5\right) \]

      3. *-commutative 99.8%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{\frac{8 \cdot \left(1 - v\right)}{\color{blue}{{\left(w \cdot r\right)}^{2} \cdot \mathsf{fma}\left(v, -2, 3\right)}}} + 4.5\right) \]

      4. times-frac 99.7%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{\color{blue}{\frac{8}{{\left(w \cdot r\right)}^{2}} \cdot \frac{1 - v}{\mathsf{fma}\left(v, -2, 3\right)}}} + 4.5\right) \]

      5. *-commutative 99.7%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{\frac{8}{{\color{blue}{\left(r \cdot w\right)}}^{2}} \cdot \frac{1 - v}{\mathsf{fma}\left(v, -2, 3\right)}} + 4.5\right) \]

   8. Simplified 99.7%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{1}{\frac{8}{{\left(r \cdot w\right)}^{2}} \cdot \frac{1 - v}{\mathsf{fma}\left(v, -2, 3\right)}}} + 4.5\right) \]

   9. Taylor expanded in v around 0 55.4%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(0.125 \cdot \left(v \cdot \left(-2 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -3 \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + 0.375 \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} + 4.5\right) \]
   10. Step-by-step derivation

       1. +-commutative 55.4%

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

       2. associate-*r* 55.4%

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

       3. distribute-rgt-out-- 70.6%

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

       4. metadata-eval 70.6%

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

       5. *-rgt-identity 70.6%

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

       6. distribute-rgt-out 83.5%

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

       7. unpow2 83.5%

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

       8. unpow2 83.5%

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

       9. swap-sqr 99.7%

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

       10. unpow2 99.7%

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

   11. Simplified 99.7%

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

   if 8.9999999999999995e-23 < v

   1. Initial program 79.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around inf 86.6%

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

      1. *-commutative 86.6%

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

      2. *-commutative 86.6%

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

      3. unpow2 86.6%

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

      4. unpow2 86.6%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.7 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{r \cdot \frac{r}{2}} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 9 \cdot 10^{-23}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left({\left(r \cdot w\right)}^{2} \cdot \left(0.375 + v \cdot 0.125\right) + 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Simplified 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-*r* 99.7%

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

   3. times-frac 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. inv-pow 99.7%

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

   2. unpow-prod-down 99.8%

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

   3. inv-pow 99.8%

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

   4. inv-pow 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Simplified 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-*r* 99.7%

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

   3. times-frac 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. associate-/r* 99.8%

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

   3. *-commutative 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 96.4% accurate, 0.2× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Simplified 98.3%

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

5. Final simplification 98.3%

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

Alternative 5: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Simplified 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-*r* 99.7%

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

   3. times-frac 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{1}{r \cdot \frac{r}{2}}\\ t_2 := -1.5 - t\_0 \cdot 0.25\\ \mathbf{if}\;v \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;t\_1 + t\_2\\ \mathbf{elif}\;v \leq 10^{-24}:\\ \;\;\;\;t\_1 + \left(-1.5 - 0.375 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w)))
        (t_1 (/ 1.0 (* r (/ r 2.0))))
        (t_2 (- -1.5 (* t_0 0.25))))
   (if (<= v -3.4e+24)
     (+ t_1 t_2)
     (if (<= v 1e-24)
       (+ t_1 (- -1.5 (* 0.375 t_0)))
       (+ (/ (/ 2.0 r) r) t_2)))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = 1.0 / (r * (r / 2.0));
	double t_2 = -1.5 - (t_0 * 0.25);
	double tmp;
	if (v <= -3.4e+24) {
		tmp = t_1 + t_2;
	} else if (v <= 1e-24) {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	} else {
		tmp = ((2.0 / r) / r) + t_2;
	}
	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 = (r * w) * (r * w)
    t_1 = 1.0d0 / (r * (r / 2.0d0))
    t_2 = (-1.5d0) - (t_0 * 0.25d0)
    if (v <= (-3.4d+24)) then
        tmp = t_1 + t_2
    else if (v <= 1d-24) then
        tmp = t_1 + ((-1.5d0) - (0.375d0 * t_0))
    else
        tmp = ((2.0d0 / r) / r) + t_2
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = 1.0 / (r * (r / 2.0));
	double t_2 = -1.5 - (t_0 * 0.25);
	double tmp;
	if (v <= -3.4e+24) {
		tmp = t_1 + t_2;
	} else if (v <= 1e-24) {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	} else {
		tmp = ((2.0 / r) / r) + t_2;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = 1.0 / (r * (r / 2.0))
	t_2 = -1.5 - (t_0 * 0.25)
	tmp = 0
	if v <= -3.4e+24:
		tmp = t_1 + t_2
	elif v <= 1e-24:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	else:
		tmp = ((2.0 / r) / r) + t_2
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(1.0 / Float64(r * Float64(r / 2.0)))
	t_2 = Float64(-1.5 - Float64(t_0 * 0.25))
	tmp = 0.0
	if (v <= -3.4e+24)
		tmp = Float64(t_1 + t_2);
	elseif (v <= 1e-24)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = 1.0 / (r * (r / 2.0));
	t_2 = -1.5 - (t_0 * 0.25);
	tmp = 0.0;
	if (v <= -3.4e+24)
		tmp = t_1 + t_2;
	elseif (v <= 1e-24)
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	else
		tmp = ((2.0 / r) / r) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(r * N[(r / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.5 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -3.4e+24], N[(t$95$1 + t$95$2), $MachinePrecision], If[LessEqual[v, 1e-24], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.4000000000000001e24

   1. Initial program 86.1%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in v around inf 91.9%

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

      1. *-commutative 91.9%

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

      2. *-commutative 91.9%

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

      3. unpow2 91.9%

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

      4. unpow2 91.9%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. clear-num 99.8%

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

       2. inv-pow 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. unpow-1 99.8%

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

       2. associate-/r/ 99.8%

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

   13. Simplified 99.8%

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

   if -3.4000000000000001e24 < v < 9.99999999999999924e-25

   1. Initial program 87.3%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in v around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. unpow2 83.4%

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

      4. unpow2 83.4%

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

      5. swap-sqr 99.1%

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

      6. unpow2 99.1%

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

      7. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. unpow2 83.7%

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

   9. Applied egg-rr 99.1%

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

       1. clear-num 83.7%

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

       2. inv-pow 83.7%

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

   11. Applied egg-rr 99.1%

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

       1. unpow-1 83.7%

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

       2. associate-/r/ 83.7%

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

   13. Simplified 99.2%

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

   if 9.99999999999999924e-25 < v

   1. Initial program 79.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around inf 86.6%

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

      1. *-commutative 86.6%

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

      2. *-commutative 86.6%

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

      3. unpow2 86.6%

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

      4. unpow2 86.6%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{r \cdot \frac{r}{2}} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 10^{-24}:\\ \;\;\;\;\frac{1}{r \cdot \frac{r}{2}} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -3.4 \cdot 10^{+24} \lor \neg \left(v \leq 9 \cdot 10^{-23}\right):\\ \;\;\;\;t\_1 + \left(-1.5 - t\_0 \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 - 0.375 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (/ (/ 2.0 r) r)))
   (if (or (<= v -3.4e+24) (not (<= v 9e-23)))
     (+ t_1 (- -1.5 (* t_0 0.25)))
     (+ t_1 (- -1.5 (* 0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double tmp;
	if ((v <= -3.4e+24) || !(v <= 9e-23)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = (2.0d0 / r) / r
    if ((v <= (-3.4d+24)) .or. (.not. (v <= 9d-23))) then
        tmp = t_1 + ((-1.5d0) - (t_0 * 0.25d0))
    else
        tmp = t_1 + ((-1.5d0) - (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double tmp;
	if ((v <= -3.4e+24) || !(v <= 9e-23)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = (2.0 / r) / r
	tmp = 0
	if (v <= -3.4e+24) or not (v <= 9e-23):
		tmp = t_1 + (-1.5 - (t_0 * 0.25))
	else:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if ((v <= -3.4e+24) || !(v <= 9e-23))
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * 0.25)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = (2.0 / r) / r;
	tmp = 0.0;
	if ((v <= -3.4e+24) || ~((v <= 9e-23)))
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	else
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[Or[LessEqual[v, -3.4e+24], N[Not[LessEqual[v, 9e-23]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.4000000000000001e24 or 8.9999999999999995e-23 < v

   1. Initial program 82.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around inf 89.1%

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

      1. *-commutative 89.1%

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

      2. *-commutative 89.1%

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

      3. unpow2 89.1%

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

      4. unpow2 89.1%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -3.4000000000000001e24 < v < 8.9999999999999995e-23

   1. Initial program 87.3%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in v around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. unpow2 83.4%

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

      4. unpow2 83.4%

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

      5. swap-sqr 99.1%

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

      6. unpow2 99.1%

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

      7. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. unpow2 83.7%

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

   9. Applied egg-rr 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.4 \cdot 10^{+24} \lor \neg \left(v \leq 9 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{\frac{2}{r}}{r}\\ t_2 := -1.5 - t\_0 \cdot 0.25\\ \mathbf{if}\;v \leq -3 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{r \cdot \frac{r}{2}} + t\_2\\ \mathbf{elif}\;v \leq 9 \cdot 10^{-23}:\\ \;\;\;\;t\_1 + \left(-1.5 - 0.375 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w)))
        (t_1 (/ (/ 2.0 r) r))
        (t_2 (- -1.5 (* t_0 0.25))))
   (if (<= v -3e+21)
     (+ (/ 1.0 (* r (/ r 2.0))) t_2)
     (if (<= v 9e-23) (+ t_1 (- -1.5 (* 0.375 t_0))) (+ t_1 t_2)))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double t_2 = -1.5 - (t_0 * 0.25);
	double tmp;
	if (v <= -3e+21) {
		tmp = (1.0 / (r * (r / 2.0))) + t_2;
	} else if (v <= 9e-23) {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	} else {
		tmp = t_1 + t_2;
	}
	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 = (r * w) * (r * w)
    t_1 = (2.0d0 / r) / r
    t_2 = (-1.5d0) - (t_0 * 0.25d0)
    if (v <= (-3d+21)) then
        tmp = (1.0d0 / (r * (r / 2.0d0))) + t_2
    else if (v <= 9d-23) then
        tmp = t_1 + ((-1.5d0) - (0.375d0 * t_0))
    else
        tmp = t_1 + t_2
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double t_2 = -1.5 - (t_0 * 0.25);
	double tmp;
	if (v <= -3e+21) {
		tmp = (1.0 / (r * (r / 2.0))) + t_2;
	} else if (v <= 9e-23) {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	} else {
		tmp = t_1 + t_2;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = (2.0 / r) / r
	t_2 = -1.5 - (t_0 * 0.25)
	tmp = 0
	if v <= -3e+21:
		tmp = (1.0 / (r * (r / 2.0))) + t_2
	elif v <= 9e-23:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	else:
		tmp = t_1 + t_2
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(Float64(2.0 / r) / r)
	t_2 = Float64(-1.5 - Float64(t_0 * 0.25))
	tmp = 0.0
	if (v <= -3e+21)
		tmp = Float64(Float64(1.0 / Float64(r * Float64(r / 2.0))) + t_2);
	elseif (v <= 9e-23)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	else
		tmp = Float64(t_1 + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = (2.0 / r) / r;
	t_2 = -1.5 - (t_0 * 0.25);
	tmp = 0.0;
	if (v <= -3e+21)
		tmp = (1.0 / (r * (r / 2.0))) + t_2;
	elseif (v <= 9e-23)
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	else
		tmp = t_1 + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, Block[{t$95$2 = N[(-1.5 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -3e+21], N[(N[(1.0 / N[(r * N[(r / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[v, 9e-23], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -3e21

   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 99.7%

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

   5. Taylor expanded in v around inf 92.2%

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

      1. *-commutative 92.2%

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

      2. *-commutative 92.2%

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

      3. unpow2 92.2%

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

      4. unpow2 92.2%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. clear-num 99.8%

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

       2. inv-pow 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. unpow-1 99.8%

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

       2. associate-/r/ 99.8%

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

   13. Simplified 99.8%

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

   if -3e21 < v < 8.9999999999999995e-23

   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 96.7%

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

   5. Taylor expanded in v around 0 83.0%

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

      1. *-commutative 83.0%

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

      2. *-commutative 83.0%

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

      3. unpow2 83.0%

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

      4. unpow2 83.0%

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

      5. swap-sqr 99.1%

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

      6. unpow2 99.1%

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

      7. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. unpow2 83.3%

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

   9. Applied egg-rr 99.1%

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

   if 8.9999999999999995e-23 < v

   1. Initial program 79.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around inf 86.6%

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

      1. *-commutative 86.6%

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

      2. *-commutative 86.6%

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

      3. unpow2 86.6%

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

      4. unpow2 86.6%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{r \cdot \frac{r}{2}} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 9 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Simplified 98.3%

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

5. Taylor expanded in v around inf 81.2%

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

   1. *-commutative 81.2%

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

   2. *-commutative 81.2%

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

   3. unpow2 81.2%

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

   4. unpow2 81.2%

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

   5. swap-sqr 91.7%

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

   6. unpow2 91.7%

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

   7. *-commutative 91.7%

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

7. Simplified 91.7%

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

   1. unpow2 91.7%

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

9. Applied egg-rr 91.7%

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

10. Final simplification 91.7%

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

Reproduce

herbie shell --seed 2024026 
(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))