Rosa's TurbineBenchmark

Percentage Accurate: 84.4% → 99.2%

Time: 11.8s

Alternatives: 6

Speedup: 1.7×

• 52.5% 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: 99.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Simplified 95.3%

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

   1. associate-*r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. associate-/l* 99.8%

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

   3. clear-num 99.8%

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

   4. associate-/r* 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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)) / (((r * w) ** (-2.0d0)) * (v + (-1.0d0))))
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) / (Math.pow((r * w), -2.0) * (v + -1.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Simplified 95.3%

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

   1. associate-*r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. /-rgt-identity 99.8%

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

   3. associate-*r/ 99.8%

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

   4. unpow2 99.8%

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

   5. clear-num 99.8%

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

   6. div-inv 99.8%

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

   7. frac-2neg 99.8%

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

   8. div-inv 99.8%

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

   9. pow-flip 99.8%

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

   10. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

   1. neg-sub0 99.8%

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

   2. fma-udef 99.8%

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

   3. *-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate--r+ 99.8%

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

   7. metadata-eval 99.8%

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

   8. *-commutative 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-rgt-neg-in 99.8%

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

   11. neg-sub0 99.8%

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

   12. associate--r- 99.8%

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

   13. metadata-eval 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Simplified 95.3%

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

   1. associate-*r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. /-rgt-identity 99.8%

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

   3. associate-*r/ 99.8%

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

   4. unpow2 99.8%

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

   5. clear-num 99.8%

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

   6. div-inv 99.8%

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

   7. frac-2neg 99.8%

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

   8. div-inv 99.8%

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

   9. pow-flip 99.8%

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

   10. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

   1. neg-sub0 99.8%

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

   2. fma-udef 99.8%

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

   3. *-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate--r+ 99.8%

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

   7. metadata-eval 99.8%

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

   8. *-commutative 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-rgt-neg-in 99.8%

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

   11. neg-sub0 99.8%

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

   12. associate--r- 99.8%

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

   13. metadata-eval 99.8%

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

9. Simplified 99.8%

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

    1. metadata-eval 99.8%

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

    2. pow-prod-up 99.7%

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

    3. inv-pow 99.7%

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

    4. inv-pow 99.7%

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

    5. frac-2neg 99.7%

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

    6. metadata-eval 99.7%

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

    7. frac-times 99.8%

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

    8. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Simplified 95.3%

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

   1. associate-*r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. /-rgt-identity 99.8%

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

   3. associate-*r/ 99.8%

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

   4. unpow2 99.8%

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

   5. clear-num 99.8%

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

   6. div-inv 99.8%

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

   7. frac-2neg 99.8%

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

   8. div-inv 99.8%

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

   9. pow-flip 99.8%

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

   10. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

   1. neg-sub0 99.8%

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

   2. fma-udef 99.8%

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

   3. *-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate--r+ 99.8%

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

   7. metadata-eval 99.8%

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

   8. *-commutative 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-rgt-neg-in 99.8%

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

   11. neg-sub0 99.8%

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

   12. associate--r- 99.8%

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

   13. metadata-eval 99.8%

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

9. Simplified 99.8%

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

    1. metadata-eval 99.8%

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

    2. pow-prod-up 99.7%

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

    3. inv-pow 99.7%

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

    4. inv-pow 99.7%

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

    5. associate-/r* 99.8%

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

    6. associate-/l/ 99.8%

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

    7. frac-times 99.8%

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

    8. div-inv 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 5: 93.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Simplified 86.6%

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

4. Taylor expanded in v around 0 80.9%

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

   1. *-commutative 80.9%

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

   2. unpow2 80.9%

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

   3. unpow2 80.9%

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

   4. swap-sqr 94.9%

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

   5. unpow2 94.9%

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

6. Simplified 94.9%

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

   1. unpow2 94.9%

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

8. Applied egg-rr 94.9%

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

   1. associate-*l* 94.9%

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

   2. add-sqr-sqrt 53.1%

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

   3. sqrt-prod 74.0%

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

   4. associate-*r* 73.5%

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

   5. sqrt-prod 37.5%

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

   6. /-rgt-identity 37.5%

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

   7. sqrt-prod 73.5%

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

   8. associate-/r/ 73.5%

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

   9. add-sqr-sqrt 52.7%

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

   10. sqrt-prod 93.7%

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

   11. associate-*r* 93.7%

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

   12. sqrt-prod 44.8%

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

   13. /-rgt-identity 44.8%

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

   14. sqrt-prod 93.7%

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

   15. associate-/r/ 93.7%

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

   16. associate-*l* 93.7%

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

10. Applied egg-rr 94.9%

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

    1. associate-*r* 94.9%

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

    2. *-un-lft-identity 94.9%

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

    3. times-frac 94.9%

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

12. Applied egg-rr 94.9%

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

13. Final simplification 94.9%

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

Alternative 6: 93.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Simplified 86.6%

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

4. Taylor expanded in v around 0 80.9%

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

   1. *-commutative 80.9%

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

   2. unpow2 80.9%

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

   3. unpow2 80.9%

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

   4. swap-sqr 94.9%

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

   5. unpow2 94.9%

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

6. Simplified 94.9%

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

   1. unpow2 94.9%

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

8. Applied egg-rr 94.9%

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

   1. associate-*l* 94.9%

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

   2. add-sqr-sqrt 53.1%

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

   3. sqrt-prod 74.0%

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

   4. associate-*r* 73.5%

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

   5. sqrt-prod 37.5%

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

   6. /-rgt-identity 37.5%

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

   7. sqrt-prod 73.5%

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

   8. associate-/r/ 73.5%

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

   9. add-sqr-sqrt 52.7%

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

   10. sqrt-prod 93.7%

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

   11. associate-*r* 93.7%

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

   12. sqrt-prod 44.8%

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

   13. /-rgt-identity 44.8%

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

   14. sqrt-prod 93.7%

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

   15. associate-/r/ 93.7%

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

   16. associate-*l* 93.7%

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

10. Applied egg-rr 94.9%

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

    1. div-inv 94.9%

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

    2. *-commutative 94.9%

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

    3. associate-/l/ 94.9%

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

    4. inv-pow 94.9%

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

    5. pow-flip 94.9%

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

    6. metadata-eval 94.9%

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

    7. pow1 94.9%

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

12. Applied egg-rr 94.9%

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

13. Final simplification 94.9%

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

Reproduce

herbie shell --seed 2023309 
(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))