Rosa's TurbineBenchmark

Percentage Accurate: 85.1% → 99.8%

Time: 12.9s

Alternatives: 7

Speedup: 1.7×

• 53.5% 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: 85.1% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right)\right)\right) + -4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (+
   3.0
   (-
    (/ 2.0 (* r r))
    (* (* r w) (* (/ (fma v -0.25 0.375) (- 1.0 v)) (* r w)))))
  -4.5))

C

double code(double v, double w, double r) {
	return (3.0 + ((2.0 / (r * r)) - ((r * w) * ((fma(v, -0.25, 0.375) / (1.0 - v)) * (r * w))))) + -4.5;
}

Julia

function code(v, w, r)
	return Float64(Float64(3.0 + Float64(Float64(2.0 / Float64(r * r)) - Float64(Float64(r * w) * Float64(Float64(fma(v, -0.25, 0.375) / Float64(1.0 - v)) * Float64(r * w))))) + -4.5)
end

Wolfram

code[v_, w_, r_] := N[(N[(3.0 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 86.8%

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

   1. associate-/r/ 86.8%

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

   2. associate-*r* 78.2%

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

   3. swap-sqr 99.7%

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

   4. associate-*r* 99.7%

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

   5. +-commutative 99.7%

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

   6. distribute-rgt-in 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-*l* 99.7%

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

   9. metadata-eval 99.7%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   10. metadata-eval 99.7%

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

   11. fma-udef 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right)\right)\right) + -4.5 \]
8. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 96.9%

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

   1. associate-*r* 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. times-frac 99.7%

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

6. Applied egg-rr 99.7%

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

   1. clear-num 99.7%

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

   2. frac-times 99.7%

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

   3. metadata-eval 99.7%

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

   4. div-inv 99.7%

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

   5. /-rgt-identity 99.7%

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

   6. *-un-lft-identity 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return -4.5 + (3.0 + ((2.0 / (r * r)) - ((0.125 * (3.0 + (v * -2.0))) / (((1.0 - v) / (r * w)) * (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 = (-4.5d0) + (3.0d0 + ((2.0d0 / (r * r)) - ((0.125d0 * (3.0d0 + (v * (-2.0d0)))) / (((1.0d0 - v) / (r * w)) * (1.0d0 / (r * w))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 86.8%

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

   1. associate-*r* 96.9%

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

   2. *-commutative 96.9%

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

   3. *-un-lft-identity 96.9%

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

   4. associate-*r* 99.7%

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

   5. times-frac 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 10^{+190}:\\ \;\;\;\;-1.5 + \left(t_0 + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 1e+190)
     (+
      -1.5
      (+ t_0 (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r (* w (* r w))))))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* w (* r 0.25)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 1e+190) {
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (w <= 1d+190) then
        tmp = (-1.5d0) + (t_0 + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r * (w * (r * w)))))
    else
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (w * (r * 0.25d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 1e+190) {
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if w <= 1e+190:
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (r * w)))))
	else:
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (w <= 1e+190)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r * Float64(w * Float64(r * w))))));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.25))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (w <= 1e+190)
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	else
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 1e+190], N[(-1.5 + N[(t$95$0 + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 1.0000000000000001e190

   1. Initial program 86.1%

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

   3. Simplified 88.8%

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

      1. add-sqr-sqrt 58.6%

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

      2. pow2 58.6%

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

      3. *-commutative 58.6%

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

      4. sqrt-prod 45.6%

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

      5. sqrt-prod 23.1%

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

      6. add-sqr-sqrt 49.7%

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

   6. Applied egg-rr 49.7%

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

      1. *-commutative 49.7%

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

      2. unpow-prod-down 45.6%

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

      3. pow2 45.6%

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

      4. add-sqr-sqrt 88.8%

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

      5. pow2 88.8%

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

      6. associate-*r* 97.4%

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

   8. Applied egg-rr 97.4%

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

   if 1.0000000000000001e190 < w

   1. Initial program 65.7%

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

   3. Simplified 65.7%

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

      1. associate-/r/ 65.7%

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

      2. associate-*r* 65.7%

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

      3. swap-sqr 99.9%

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

      4. associate-*r* 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*l* 99.9%

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

      9. metadata-eval 99.9%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.9%

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

      11. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around inf 99.9%

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

      1. associate-*r* 99.9%

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

   9. Simplified 99.9%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\left(0.25 \cdot r\right) \cdot w\right)} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 10^{+190}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -12500000 \lor \neg \left(v \leq 0.15\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -12500000.0) (not (<= v 0.15)))
   (+ -4.5 (+ 3.0 (- (/ 2.0 (* r r)) (* (* r w) (* w (* r 0.25))))))
   (+ -1.5 (+ (/ (/ 2.0 r) r) (* -0.375 (* (* r w) (* r w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -12500000.0) || !(v <= 0.15)) {
		tmp = -4.5 + (3.0 + ((2.0 / (r * r)) - ((r * w) * (w * (r * 0.25)))));
	} else {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if ((v <= (-12500000.0d0)) .or. (.not. (v <= 0.15d0))) then
        tmp = (-4.5d0) + (3.0d0 + ((2.0d0 / (r * r)) - ((r * w) * (w * (r * 0.25d0)))))
    else
        tmp = (-1.5d0) + (((2.0d0 / r) / r) + ((-0.375d0) * ((r * w) * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -12500000.0) || !(v <= 0.15)) {
		tmp = -4.5 + (3.0 + ((2.0 / (r * r)) - ((r * w) * (w * (r * 0.25)))));
	} else {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -12500000.0) or not (v <= 0.15):
		tmp = -4.5 + (3.0 + ((2.0 / (r * r)) - ((r * w) * (w * (r * 0.25)))))
	else:
		tmp = -1.5 + (((2.0 / r) / r) + (-0.375 * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -12500000.0) || !(v <= 0.15))
		tmp = Float64(-4.5 + Float64(3.0 + Float64(Float64(2.0 / Float64(r * r)) - Float64(Float64(r * w) * Float64(w * Float64(r * 0.25))))));
	else
		tmp = Float64(-1.5 + Float64(Float64(Float64(2.0 / r) / r) + Float64(-0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -12500000.0) || ~((v <= 0.15)))
		tmp = -4.5 + (3.0 + ((2.0 / (r * r)) - ((r * w) * (w * (r * 0.25)))));
	else
		tmp = -1.5 + (((2.0 / r) / r) + (-0.375 * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -12500000.0], N[Not[LessEqual[v, 0.15]], $MachinePrecision]], N[(-4.5 + N[(3.0 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.25e7 or 0.149999999999999994 < v

   1. Initial program 83.2%

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

   3. Simplified 87.6%

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

      1. associate-/r/ 87.6%

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

      2. associate-*r* 80.1%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.8%

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

      11. fma-udef 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf 99.6%

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

      1. associate-*r* 99.6%

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

   9. Simplified 99.6%

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

   if -1.25e7 < v < 0.149999999999999994

   1. Initial program 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in v around 0 75.6%

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

      1. *-commutative 75.6%

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

      2. unpow2 75.6%

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

      3. unpow2 75.6%

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

      4. swap-sqr 98.6%

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

      5. unpow2 98.6%

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

   7. Simplified 98.6%

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

      1. unpow2 98.6%

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

   9. Applied egg-rr 98.6%

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

       1. associate-/r* 98.6%

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

       2. div-inv 98.6%

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

       3. *-un-lft-identity 98.6%

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

       4. times-frac 98.6%

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

       5. metadata-eval 98.6%

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

   11. Applied egg-rr 98.6%

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

       1. associate-*r/ 98.6%

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

       2. un-div-inv 98.6%

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

   13. Applied egg-rr 98.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -12500000 \lor \neg \left(v \leq 0.15\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -12500000 \lor \neg \left(v \leq 0.15\right):\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -12500000.0) (not (<= v 0.15)))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* w (* r 0.25))))))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* w (* r 0.375)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -12500000.0) || !(v <= 0.15)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-12500000.0d0)) .or. (.not. (v <= 0.15d0))) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (w * (r * 0.25d0)))))
    else
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (w * (r * 0.375d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -12500000.0) || !(v <= 0.15)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -12500000.0) or not (v <= 0.15):
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))))
	else:
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -12500000.0) || !(v <= 0.15))
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.25))))));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -12500000.0) || ~((v <= 0.15)))
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	else
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -12500000.0], N[Not[LessEqual[v, 0.15]], $MachinePrecision]], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.25e7 or 0.149999999999999994 < v

   1. Initial program 83.2%

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

   3. Simplified 87.6%

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

      1. associate-/r/ 87.6%

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

      2. associate-*r* 80.1%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.8%

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

      11. fma-udef 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf 99.6%

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

      1. associate-*r* 99.6%

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

   9. Simplified 99.6%

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

   if -1.25e7 < v < 0.149999999999999994

   1. Initial program 85.6%

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

   3. Simplified 85.6%

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

      1. associate-/r/ 85.6%

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

      2. associate-*r* 75.6%

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

      3. swap-sqr 99.6%

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

      4. associate-*r* 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-rgt-in 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*l* 99.7%

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

      9. metadata-eval 99.7%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.7%

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

      11. fma-udef 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around 0 98.6%

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

      1. associate-*r* 98.7%

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

   9. Simplified 98.7%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\left(0.375 \cdot r\right) \cdot w\right)} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -12500000 \lor \neg \left(v \leq 0.15\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 86.8%

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

5. Taylor expanded in v around 0 74.2%

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

   1. *-commutative 74.2%

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

   2. unpow2 74.2%

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

   3. unpow2 74.2%

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

   4. swap-sqr 89.4%

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

   5. unpow2 89.4%

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

7. Simplified 89.4%

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

   1. unpow2 89.4%

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

9. Applied egg-rr 89.4%

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

10. Final simplification 89.4%

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

Reproduce

herbie shell --seed 2024019 
(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))