Rosa's TurbineBenchmark

Percentage Accurate: 84.4% → 97.4%

Time: 13.1s

Alternatives: 5

Speedup: 1.7×

• 52.9% 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: 84.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5400000000 \lor \neg \left(v \leq 2 \cdot 10^{-47}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -5400000000.0) (not (<= v 2e-47)))
   (+ -1.5 (+ (/ 2.0 (* r r)) (* (* r (* w (* r w))) -0.25)))
   (+ (/ (/ 2.0 r) r) (- -1.5 (* (* r w) (* r (* w 0.375)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -5400000000.0) || !(v <= 2e-47)) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * (r * w))) * -0.25));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - ((r * w) * (r * (w * 0.375))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if ((v <= (-5400000000.0d0)) .or. (.not. (v <= 2d-47))) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((r * (w * (r * w))) * (-0.25d0)))
    else
        tmp = ((2.0d0 / r) / r) + ((-1.5d0) - ((r * w) * (r * (w * 0.375d0))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -5400000000.0) || !(v <= 2e-47)) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * (r * w))) * -0.25));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - ((r * w) * (r * (w * 0.375))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -5400000000.0) or not (v <= 2e-47):
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * (r * w))) * -0.25))
	else:
		tmp = ((2.0 / r) / r) + (-1.5 - ((r * w) * (r * (w * 0.375))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -5400000000.0) || !(v <= 2e-47))
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(r * Float64(w * Float64(r * w))) * -0.25)));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(Float64(r * w) * Float64(r * Float64(w * 0.375)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -5400000000.0) || ~((v <= 2e-47)))
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * (r * w))) * -0.25));
	else
		tmp = ((2.0 / r) / r) + (-1.5 - ((r * w) * (r * (w * 0.375))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -5400000000.0], N[Not[LessEqual[v, 2e-47]], $MachinePrecision]], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -5.4e9 or 1.9999999999999999e-47 < v

   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 83.7%

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

   5. Taylor expanded in v around inf 86.1%

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

      1. *-commutative 86.1%

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

      2. *-commutative 86.1%

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

      3. unpow2 86.1%

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

      4. unpow2 86.1%

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

      5. swap-sqr 99.7%

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

      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%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -5.4e9 < v < 1.9999999999999999e-47

   1. Initial program 83.7%

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

   3. Simplified 96.5%

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

      1. frac-2neg 96.5%

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

      2. *-commutative 96.5%

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

      3. associate-*r* 83.7%

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

      4. div-inv 83.7%

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

      5. associate-*r* 96.5%

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

      6. *-commutative 96.5%

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

      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. associate-/r/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r* 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in v around 0 99.8%

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

       1. associate-*r* 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in r around 0 99.8%

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

       1. *-commutative 99.8%

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

       2. associate-*r* 99.9%

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

   16. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5400000000 \lor \neg \left(v \leq 2 \cdot 10^{-47}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2250000000 \lor \neg \left(v \leq 1.2 \cdot 10^{-46}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -2250000000.0) (not (<= v 1.2e-46)))
   (+ -1.5 (+ (/ 2.0 (* r r)) (* (* r (* w (* r w))) -0.25)))
   (+ (/ (/ 2.0 r) r) (- -1.5 (* (* r w) (* (* r w) 0.375))))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -2250000000.0) || !(v <= 1.2e-46)) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * (r * w))) * -0.25));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - ((r * w) * ((r * w) * 0.375)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if ((v <= (-2250000000.0d0)) .or. (.not. (v <= 1.2d-46))) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((r * (w * (r * w))) * (-0.25d0)))
    else
        tmp = ((2.0d0 / r) / r) + ((-1.5d0) - ((r * w) * ((r * w) * 0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -2250000000.0) || !(v <= 1.2e-46)) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * (r * w))) * -0.25));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - ((r * w) * ((r * w) * 0.375)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -2250000000.0) or not (v <= 1.2e-46):
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * (r * w))) * -0.25))
	else:
		tmp = ((2.0 / r) / r) + (-1.5 - ((r * w) * ((r * w) * 0.375)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -2250000000.0) || !(v <= 1.2e-46))
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(r * Float64(w * Float64(r * w))) * -0.25)));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(Float64(r * w) * Float64(Float64(r * w) * 0.375))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -2250000000.0) || ~((v <= 1.2e-46)))
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * (r * w))) * -0.25));
	else
		tmp = ((2.0 / r) / r) + (-1.5 - ((r * w) * ((r * w) * 0.375)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -2250000000.0], N[Not[LessEqual[v, 1.2e-46]], $MachinePrecision]], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.25e9 or 1.20000000000000007e-46 < v

   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 83.7%

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

   5. Taylor expanded in v around inf 86.1%

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

      1. *-commutative 86.1%

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

      2. *-commutative 86.1%

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

      3. unpow2 86.1%

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

      4. unpow2 86.1%

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

      5. swap-sqr 99.7%

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

      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%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -2.25e9 < v < 1.20000000000000007e-46

   1. Initial program 83.7%

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

   3. Simplified 96.5%

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

      1. frac-2neg 96.5%

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

      2. *-commutative 96.5%

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

      3. associate-*r* 83.7%

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

      4. div-inv 83.7%

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

      5. associate-*r* 96.5%

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

      6. *-commutative 96.5%

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

      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. associate-/r/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r* 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in v around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2250000000 \lor \neg \left(v \leq 1.2 \cdot 10^{-46}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.375\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Simplified 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. frac-2neg 98.3%

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

   2. *-commutative 98.3%

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

   3. associate-*r* 87.2%

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

   4. div-inv 87.2%

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

   5. associate-*r* 98.3%

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

   6. *-commutative 98.3%

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

   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. associate-/r/ 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*r* 99.8%

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

   4. sub-neg 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. metadata-eval 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 93.3% accurate, 1.7× speedup

Math

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

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(2.0 / Float64(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[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

3. Simplified 81.4%

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

5. Taylor expanded in v around inf 80.4%

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

   1. *-commutative 80.4%

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

   2. *-commutative 80.4%

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

   3. unpow2 80.4%

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

   4. unpow2 80.4%

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

   5. swap-sqr 95.1%

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

   6. unpow2 95.1%

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

   7. *-commutative 95.1%

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

7. Simplified 95.1%

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

   1. unpow2 95.1%

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

9. Applied egg-rr 95.1%

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

10. Final simplification 95.1%

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

Alternative 5: 57.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Simplified 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. frac-2neg 98.3%

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

   2. *-commutative 98.3%

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

   3. associate-*r* 87.2%

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

   4. div-inv 87.2%

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

   5. associate-*r* 98.3%

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

   6. *-commutative 98.3%

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

   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. associate-/r/ 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*r* 99.8%

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

   4. sub-neg 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. metadata-eval 99.8%

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

10. Applied egg-rr 99.8%

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

11. Taylor expanded in r around 0 56.5%

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

12. Final simplification 56.5%

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

Reproduce

herbie shell --seed 2024031 
(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))