Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 99.7%

Time: 19.9s

Alternatives: 5

Speedup: 1.7×

• 53.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ t_1 := -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{if}\;v \leq -5 \cdot 10^{+66}:\\ \;\;\;\;-1.5 + \left(t_0 + t_1\right)\\ \mathbf{elif}\;v \leq 7.6 \cdot 10^{+15}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r \cdot \left(-0.375 + v \cdot 0.25\right)}{\frac{v + -1}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_1 + \frac{2}{r \cdot r}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)) (t_1 (* -0.25 (* (* r w) (* r w)))))
   (if (<= v -5e+66)
     (+ -1.5 (+ t_0 t_1))
     (if (<= v 7.6e+15)
       (+
        t_0
        (- -1.5 (* (* r w) (/ (* r (+ -0.375 (* v 0.25))) (/ (+ v -1.0) w)))))
       (+ -1.5 (+ t_1 (/ 2.0 (* r r))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double t_1 = -0.25 * ((r * w) * (r * w));
	double tmp;
	if (v <= -5e+66) {
		tmp = -1.5 + (t_0 + t_1);
	} else if (v <= 7.6e+15) {
		tmp = t_0 + (-1.5 - ((r * w) * ((r * (-0.375 + (v * 0.25))) / ((v + -1.0) / w))));
	} else {
		tmp = -1.5 + (t_1 + (2.0 / (r * r)));
	}
	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 = (-0.25d0) * ((r * w) * (r * w))
    if (v <= (-5d+66)) then
        tmp = (-1.5d0) + (t_0 + t_1)
    else if (v <= 7.6d+15) then
        tmp = t_0 + ((-1.5d0) - ((r * w) * ((r * ((-0.375d0) + (v * 0.25d0))) / ((v + (-1.0d0)) / w))))
    else
        tmp = (-1.5d0) + (t_1 + (2.0d0 / (r * r)))
    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 = -0.25 * ((r * w) * (r * w));
	double tmp;
	if (v <= -5e+66) {
		tmp = -1.5 + (t_0 + t_1);
	} else if (v <= 7.6e+15) {
		tmp = t_0 + (-1.5 - ((r * w) * ((r * (-0.375 + (v * 0.25))) / ((v + -1.0) / w))));
	} else {
		tmp = -1.5 + (t_1 + (2.0 / (r * r)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	t_1 = -0.25 * ((r * w) * (r * w))
	tmp = 0
	if v <= -5e+66:
		tmp = -1.5 + (t_0 + t_1)
	elif v <= 7.6e+15:
		tmp = t_0 + (-1.5 - ((r * w) * ((r * (-0.375 + (v * 0.25))) / ((v + -1.0) / w))))
	else:
		tmp = -1.5 + (t_1 + (2.0 / (r * r)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	t_1 = Float64(-0.25 * Float64(Float64(r * w) * Float64(r * w)))
	tmp = 0.0
	if (v <= -5e+66)
		tmp = Float64(-1.5 + Float64(t_0 + t_1));
	elseif (v <= 7.6e+15)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(Float64(r * Float64(-0.375 + Float64(v * 0.25))) / Float64(Float64(v + -1.0) / w)))));
	else
		tmp = Float64(-1.5 + Float64(t_1 + Float64(2.0 / Float64(r * r))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / r) / r;
	t_1 = -0.25 * ((r * w) * (r * w));
	tmp = 0.0;
	if (v <= -5e+66)
		tmp = -1.5 + (t_0 + t_1);
	elseif (v <= 7.6e+15)
		tmp = t_0 + (-1.5 - ((r * w) * ((r * (-0.375 + (v * 0.25))) / ((v + -1.0) / w))));
	else
		tmp = -1.5 + (t_1 + (2.0 / (r * r)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, Block[{t$95$1 = N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -5e+66], N[(-1.5 + N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 7.6e+15], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(N[(r * N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$1 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.99999999999999991e66

   1. Initial program 78.8%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in v around inf 83.0%

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

      1. *-commutative 83.0%

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

      2. *-commutative 83.0%

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

      3. unpow2 83.0%

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

      4. unpow2 83.0%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-/r* 99.8%

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

       2. div-inv 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. associate-*r/ 99.8%

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

       2. *-rgt-identity 99.8%

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

   13. Simplified 99.8%

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

   if -4.99999999999999991e66 < v < 7.6e15

   1. Initial program 87.7%

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

   3. Simplified 97.6%

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

      1. frac-2neg 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-*r* 87.8%

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

      4. div-inv 87.7%

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

      5. associate-*r* 97.6%

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

      6. *-commutative 97.6%

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

      7. associate-*r* 99.8%

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

      8. pow2 99.8%

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

      9. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. neg-sub0 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      4. fma-udef 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      5. *-commutative 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      7. associate--r+ 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-neg-frac 99.8%

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

      11. neg-sub0 99.8%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{0 - \left(1 - v\right)}}{{\left(w \cdot r\right)}^{2}}}\right) \]

      12. associate--r- 99.8%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{\left(0 - 1\right) + v}}{{\left(w \cdot r\right)}^{2}}}\right) \]

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. expm1-log1p-u 40.6%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 + v}{{\left(r \cdot w\right)}^{2}}\right)\right)}}\right) \]

      2. expm1-udef 5.0%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\color{blue}{e^{\mathsf{log1p}\left(\frac{-1 + v}{{\left(r \cdot w\right)}^{2}}\right)} - 1}}\right) \]

      3. div-inv 5.0%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{e^{\mathsf{log1p}\left(\color{blue}{\left(-1 + v\right) \cdot \frac{1}{{\left(r \cdot w\right)}^{2}}}\right)} - 1}\right) \]

      4. pow-flip 5.0%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{e^{\mathsf{log1p}\left(\left(-1 + v\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(-2\right)}}\right)} - 1}\right) \]

      5. metadata-eval 5.0%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{e^{\mathsf{log1p}\left(\left(-1 + v\right) \cdot {\left(r \cdot w\right)}^{\color{blue}{-2}}\right)} - 1}\right) \]

   10. Applied egg-rr 5.0%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\color{blue}{e^{\mathsf{log1p}\left(\left(-1 + v\right) \cdot {\left(r \cdot w\right)}^{-2}\right)} - 1}}\right) \]
   11. Step-by-step derivation

       1. expm1-def 40.5%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-1 + v\right) \cdot {\left(r \cdot w\right)}^{-2}\right)\right)}}\right) \]

       2. expm1-log1p 99.8%

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

   12. Simplified 99.8%

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

       1. metadata-eval 99.8%

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

       2. pow-flip 99.8%

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

       3. pow2 99.8%

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

       4. div-inv 99.8%

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

       5. associate-/r* 99.8%

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

   14. Applied egg-rr 99.8%

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

       1. associate-/r/ 99.8%

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

       2. *-commutative 99.8%

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

       3. div-inv 99.8%

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

       4. sub-neg 99.8%

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

       5. distribute-rgt-neg-in 99.8%

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

       6. metadata-eval 99.8%

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

       7. clear-num 99.8%

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

       8. associate-/l* 99.9%

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

   16. Applied egg-rr 99.9%

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

       1. associate-*r/ 99.9%

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

   18. Simplified 99.9%

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

   if 7.6e15 < v

   1. Initial program 79.1%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in v around inf 88.8%

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

      1. *-commutative 88.8%

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

      2. *-commutative 88.8%

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

      3. unpow2 88.8%

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

      4. unpow2 88.8%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+66}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 7.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r \cdot \left(-0.375 + v \cdot 0.25\right)}{\frac{v + -1}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ t_1 := -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{if}\;v \leq -4 \cdot 10^{+41}:\\ \;\;\;\;-1.5 + \left(t_0 + t_1\right)\\ \mathbf{elif}\;v \leq 8.5 \cdot 10^{-19}:\\ \;\;\;\;t_0 + \left(-1.5 + \frac{v \cdot -0.25 - -0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_1 + \frac{2}{r \cdot r}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)) (t_1 (* -0.25 (* (* r w) (* r w)))))
   (if (<= v -4e+41)
     (+ -1.5 (+ t_0 t_1))
     (if (<= v 8.5e-19)
       (+ t_0 (+ -1.5 (/ (- (* v -0.25) -0.375) (/ (/ (/ -1.0 r) w) (* r w)))))
       (+ -1.5 (+ t_1 (/ 2.0 (* r r))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double t_1 = -0.25 * ((r * w) * (r * w));
	double tmp;
	if (v <= -4e+41) {
		tmp = -1.5 + (t_0 + t_1);
	} else if (v <= 8.5e-19) {
		tmp = t_0 + (-1.5 + (((v * -0.25) - -0.375) / (((-1.0 / r) / w) / (r * w))));
	} else {
		tmp = -1.5 + (t_1 + (2.0 / (r * r)));
	}
	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 = (-0.25d0) * ((r * w) * (r * w))
    if (v <= (-4d+41)) then
        tmp = (-1.5d0) + (t_0 + t_1)
    else if (v <= 8.5d-19) then
        tmp = t_0 + ((-1.5d0) + (((v * (-0.25d0)) - (-0.375d0)) / ((((-1.0d0) / r) / w) / (r * w))))
    else
        tmp = (-1.5d0) + (t_1 + (2.0d0 / (r * r)))
    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 = -0.25 * ((r * w) * (r * w));
	double tmp;
	if (v <= -4e+41) {
		tmp = -1.5 + (t_0 + t_1);
	} else if (v <= 8.5e-19) {
		tmp = t_0 + (-1.5 + (((v * -0.25) - -0.375) / (((-1.0 / r) / w) / (r * w))));
	} else {
		tmp = -1.5 + (t_1 + (2.0 / (r * r)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	t_1 = -0.25 * ((r * w) * (r * w))
	tmp = 0
	if v <= -4e+41:
		tmp = -1.5 + (t_0 + t_1)
	elif v <= 8.5e-19:
		tmp = t_0 + (-1.5 + (((v * -0.25) - -0.375) / (((-1.0 / r) / w) / (r * w))))
	else:
		tmp = -1.5 + (t_1 + (2.0 / (r * r)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	t_1 = Float64(-0.25 * Float64(Float64(r * w) * Float64(r * w)))
	tmp = 0.0
	if (v <= -4e+41)
		tmp = Float64(-1.5 + Float64(t_0 + t_1));
	elseif (v <= 8.5e-19)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(Float64(v * -0.25) - -0.375) / Float64(Float64(Float64(-1.0 / r) / w) / Float64(r * w)))));
	else
		tmp = Float64(-1.5 + Float64(t_1 + Float64(2.0 / Float64(r * r))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / r) / r;
	t_1 = -0.25 * ((r * w) * (r * w));
	tmp = 0.0;
	if (v <= -4e+41)
		tmp = -1.5 + (t_0 + t_1);
	elseif (v <= 8.5e-19)
		tmp = t_0 + (-1.5 + (((v * -0.25) - -0.375) / (((-1.0 / r) / w) / (r * w))));
	else
		tmp = -1.5 + (t_1 + (2.0 / (r * r)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, Block[{t$95$1 = N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -4e+41], N[(-1.5 + N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 8.5e-19], N[(t$95$0 + N[(-1.5 + N[(N[(N[(v * -0.25), $MachinePrecision] - -0.375), $MachinePrecision] / N[(N[(N[(-1.0 / r), $MachinePrecision] / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$1 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.00000000000000002e41

   1. Initial program 80.2%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in v around inf 84.1%

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

      1. *-commutative 84.1%

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

      2. *-commutative 84.1%

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

      3. unpow2 84.1%

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

      4. unpow2 84.1%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-/r* 99.8%

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

       2. div-inv 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. associate-*r/ 99.8%

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

       2. *-rgt-identity 99.8%

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

   13. Simplified 99.8%

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

   if -4.00000000000000002e41 < v < 8.50000000000000003e-19

   1. Initial program 87.2%

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

   3. Simplified 97.5%

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

      1. frac-2neg 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*r* 87.3%

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

      4. div-inv 87.3%

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

      5. associate-*r* 97.4%

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

      6. *-commutative 97.4%

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

      7. associate-*r* 99.8%

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

      8. pow2 99.8%

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

      9. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. neg-sub0 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      4. fma-udef 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      5. *-commutative 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      7. associate--r+ 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-neg-frac 99.8%

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

      11. neg-sub0 99.8%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{0 - \left(1 - v\right)}}{{\left(w \cdot r\right)}^{2}}}\right) \]

      12. associate--r- 99.8%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{\left(0 - 1\right) + v}}{{\left(w \cdot r\right)}^{2}}}\right) \]

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. expm1-log1p-u 40.0%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 + v}{{\left(r \cdot w\right)}^{2}}\right)\right)}}\right) \]

      2. expm1-udef 4.5%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\color{blue}{e^{\mathsf{log1p}\left(\frac{-1 + v}{{\left(r \cdot w\right)}^{2}}\right)} - 1}}\right) \]

      3. div-inv 4.5%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{e^{\mathsf{log1p}\left(\color{blue}{\left(-1 + v\right) \cdot \frac{1}{{\left(r \cdot w\right)}^{2}}}\right)} - 1}\right) \]

      4. pow-flip 4.5%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{e^{\mathsf{log1p}\left(\left(-1 + v\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(-2\right)}}\right)} - 1}\right) \]

      5. metadata-eval 4.5%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{e^{\mathsf{log1p}\left(\left(-1 + v\right) \cdot {\left(r \cdot w\right)}^{\color{blue}{-2}}\right)} - 1}\right) \]

   10. Applied egg-rr 4.5%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\color{blue}{e^{\mathsf{log1p}\left(\left(-1 + v\right) \cdot {\left(r \cdot w\right)}^{-2}\right)} - 1}}\right) \]
   11. Step-by-step derivation

       1. expm1-def 40.0%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-1 + v\right) \cdot {\left(r \cdot w\right)}^{-2}\right)\right)}}\right) \]

       2. expm1-log1p 99.8%

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

   12. Simplified 99.8%

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

       1. metadata-eval 99.8%

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

       2. pow-flip 99.8%

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

       3. pow2 99.8%

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

       4. div-inv 99.8%

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

       5. associate-/r* 99.8%

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

   14. Applied egg-rr 99.8%

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

   15. Taylor expanded in v around 0 99.8%

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

       1. associate-/r* 99.8%

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

   17. Simplified 99.8%

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

   if 8.50000000000000003e-19 < v

   1. Initial program 79.7%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in v around inf 88.0%

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

      1. *-commutative 88.0%

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

      2. *-commutative 88.0%

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

      3. unpow2 88.0%

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

      4. unpow2 88.0%

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

      5. swap-sqr 99.7%

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

      6. unpow2 99.7%

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

      7. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. unpow2 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{+41}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 8.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \frac{v \cdot -0.25 - -0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ (/ 2.0 r) r)
  (+ -1.5 (/ (- (* v -0.25) -0.375) (/ (+ v -1.0) (* (* 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) / ((v + -1.0) / ((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)) / ((v + (-1.0d0)) / ((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) / ((v + -1.0) / ((r * w) * (r * w)))));
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

3. Simplified 97.9%

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

   1. frac-2neg 97.9%

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

   2. *-commutative 97.9%

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

   3. associate-*r* 89.1%

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

   4. div-inv 89.0%

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

   5. associate-*r* 97.9%

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

   6. *-commutative 97.9%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   4. fma-udef 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   5. *-commutative 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   6. +-commutative 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   7. associate--r+ 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{0 - \left(1 - v\right)}}{{\left(w \cdot r\right)}^{2}}}\right) \]

   12. associate--r- 99.8%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{\left(0 - 1\right) + v}}{{\left(w \cdot r\right)}^{2}}}\right) \]

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. unpow2 94.5%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 93.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

3. Simplified 82.4%

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

5. Taylor expanded in v around inf 81.6%

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

   1. *-commutative 81.6%

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

   2. *-commutative 81.6%

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

   3. unpow2 81.6%

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

   4. unpow2 81.6%

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

   5. swap-sqr 94.5%

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

   6. unpow2 94.5%

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

   7. *-commutative 94.5%

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

7. Simplified 94.5%

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

   1. unpow2 94.5%

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

9. Applied egg-rr 94.5%

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

10. Final simplification 94.5%

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

Alternative 5: 93.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return -1.5 + (((2.0 / r) / r) + (-0.25 * ((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 = (-1.5d0) + (((2.0d0 / r) / r) + ((-0.25d0) * ((r * w) * (r * w))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

3. Simplified 82.4%

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

5. Taylor expanded in v around inf 81.6%

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

   1. *-commutative 81.6%

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

   2. *-commutative 81.6%

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

   3. unpow2 81.6%

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

   4. unpow2 81.6%

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

   5. swap-sqr 94.5%

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

   6. unpow2 94.5%

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

   7. *-commutative 94.5%

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

7. Simplified 94.5%

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

   1. unpow2 94.5%

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

9. Applied egg-rr 94.5%

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

    1. associate-/r* 94.5%

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

    2. div-inv 94.4%

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

11. Applied egg-rr 94.4%

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

    1. associate-*r/ 94.5%

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

    2. *-rgt-identity 94.5%

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

13. Simplified 94.5%

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

14. Final simplification 94.5%

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

Reproduce

herbie shell --seed 2024021 
(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))