Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.7%

Time: 12.9s

Alternatives: 7

Speedup: 1.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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(r * w) * Float64(-1.0 / Float64(1.0 - v)))))))
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[(r * w), $MachinePrecision] * N[(-1.0 / N[(1.0 - v), $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 96.4%

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

   1. div-inv 96.4%

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

   2. associate-*r* 99.8%

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

   3. associate-*l* 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 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 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* 96.0%

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

   2. *-commutative 96.0%

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

   3. *-un-lft-identity 96.0%

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

   4. associate-*r* 99.4%

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

   5. times-frac 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

function tmp = code(v, w, r)
	tmp = (3.0 + ((2.0 / (r * r)) - ((0.125 * (3.0 + (v * -2.0))) / (((1.0 / w) / r) * ((1.0 - v) / (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[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 / w), $MachinePrecision] / r), $MachinePrecision] * N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $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 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* 96.0%

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

   2. *-commutative 96.0%

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

   3. *-un-lft-identity 96.0%

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

   4. associate-*r* 99.4%

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

   5. times-frac 99.4%

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

6. Applied egg-rr 99.4%

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

   1. clear-num 99.4%

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

   2. associate-/r/ 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/r* 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 4: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (-4.5d0) + (3.0d0 + ((2.0d0 / (r * r)) + (((r * w) * ((v * (-0.25d0)) - (-0.375d0))) / ((v + (-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)) + (((r * w) * ((v * -0.25) - -0.375)) / ((v + -1.0) / (r * w)))));
}

Python

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

MATLAB

function tmp = code(v, w, r)
	tmp = -4.5 + (3.0 + ((2.0 / (r * r)) + (((r * w) * ((v * -0.25) - -0.375)) / ((v + -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[(N[(r * w), $MachinePrecision] * N[(N[(v * -0.25), $MachinePrecision] - -0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(v + -1.0), $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 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* 96.0%

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

   2. *-commutative 96.0%

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

   3. *-un-lft-identity 96.0%

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

   4. associate-*r* 99.4%

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

   5. times-frac 99.4%

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

6. Applied egg-rr 99.4%

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

   1. associate-/r* 98.7%

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

   2. frac-2neg 98.7%

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

   3. associate-/r/ 98.7%

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

   4. div-inv 98.7%

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

   5. +-commutative 98.7%

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

   6. distribute-rgt-in 98.7%

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

   7. *-commutative 98.7%

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

   8. associate-*l* 99.1%

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

   9. metadata-eval 99.1%

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

   10. metadata-eval 99.1%

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

   11. fma-udef 99.1%

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

   12. clear-num 99.1%

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

   13. /-rgt-identity 99.1%

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

   14. distribute-rgt-neg-in 99.1%

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

8. Applied egg-rr 99.1%

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

   1. associate-/l* 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. associate-/r* 99.4%

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

   4. associate-*l/ 98.7%

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

   5. distribute-rgt-neg-out 98.7%

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

   6. distribute-rgt-neg-in 98.7%

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

   7. *-commutative 98.7%

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

   8. distribute-rgt-neg-in 98.7%

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

   9. neg-sub0 98.7%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{\left(r \cdot w\right) \cdot \color{blue}{\left(0 - \mathsf{fma}\left(v, -0.25, 0.375\right)\right)}}{-\frac{\frac{1 - v}{r}}{w}}\right)\right) + -4.5 \]

   10. fma-udef 98.7%

       \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{\left(r \cdot w\right) \cdot \left(0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}\right)}{-\frac{\frac{1 - v}{r}}{w}}\right)\right) + -4.5 \]

   11. *-commutative 98.7%

       \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{\left(r \cdot w\right) \cdot \left(0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)\right)}{-\frac{\frac{1 - v}{r}}{w}}\right)\right) + -4.5 \]

   12. +-commutative 98.7%

       \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{\left(r \cdot w\right) \cdot \left(0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}\right)}{-\frac{\frac{1 - v}{r}}{w}}\right)\right) + -4.5 \]

   13. *-commutative 98.7%

       \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{\left(r \cdot w\right) \cdot \left(0 - \left(0.375 + \color{blue}{v \cdot -0.25}\right)\right)}{-\frac{\frac{1 - v}{r}}{w}}\right)\right) + -4.5 \]

   14. associate--r+ 98.7%

       \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{\left(r \cdot w\right) \cdot \color{blue}{\left(\left(0 - 0.375\right) - v \cdot -0.25\right)}}{-\frac{\frac{1 - v}{r}}{w}}\right)\right) + -4.5 \]

   15. metadata-eval 98.7%

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

   16. associate-/r* 99.1%

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

   17. distribute-frac-neg 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

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

Alternative 5: 95.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 40000000000:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= v 40000000000.0)
   (+ -1.5 (+ (/ 2.0 (* r r)) (* -0.375 (* (* r w) (* r w)))))
   (+ (/ (/ 2.0 r) r) (- -1.5 (* 0.25 (* w (* r (* r w))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (v <= 40000000000.0) {
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * ((r * w) * (r * w))));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * (w * (r * (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 <= 40000000000.0d0) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((-0.375d0) * ((r * w) * (r * w))))
    else
        tmp = ((2.0d0 / r) / r) + ((-1.5d0) - (0.25d0 * (w * (r * (r * w)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (v <= 40000000000.0) {
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * ((r * w) * (r * w))));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * (w * (r * (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if v <= 40000000000.0:
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * ((r * w) * (r * w))))
	else:
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * (w * (r * (r * w)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (v <= 40000000000.0)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(-0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(0.25 * Float64(w * Float64(r * Float64(r * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (v <= 40000000000.0)
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * ((r * w) * (r * w))));
	else
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * (w * (r * (r * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[v, 40000000000.0], 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], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 4e10

   1. Initial program 87.6%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in v around 0 83.9%

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

      1. *-commutative 83.9%

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

      2. unpow2 83.9%

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

      3. unpow2 83.9%

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

      4. swap-sqr 97.5%

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

      5. unpow2 97.5%

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

   7. Simplified 97.5%

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

      1. unpow2 97.5%

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

   9. Applied egg-rr 97.5%

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

   if 4e10 < v

   1. Initial program 71.8%

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

   3. Simplified 96.4%

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

      1. div-inv 96.3%

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

      2. associate-*r* 99.7%

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

      3. associate-*l* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf 65.9%

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

      1. unpow2 65.9%

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

      2. unpow2 65.9%

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

      3. swap-sqr 99.7%

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

      4. unpow2 99.7%

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

   9. Simplified 99.7%

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

       1. unpow2 99.7%

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

       2. associate-*r* 94.7%

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

   11. Applied egg-rr 94.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 40000000000:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 190000000000:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= v 190000000000.0)
   (+ -1.5 (+ (/ 2.0 (* r r)) (* -0.375 (* (* r w) (* r w)))))
   (+ (/ (/ 2.0 r) r) (- -1.5 (* 0.25 (* r (* w (* r w))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (v <= 190000000000.0) {
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * ((r * w) * (r * w))));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * (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 <= 190000000000.0d0) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((-0.375d0) * ((r * w) * (r * w))))
    else
        tmp = ((2.0d0 / r) / r) + ((-1.5d0) - (0.25d0 * (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 <= 190000000000.0) {
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * ((r * w) * (r * w))));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * (r * (w * (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if v <= 190000000000.0:
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * ((r * w) * (r * w))))
	else:
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * (r * (w * (r * w)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (v <= 190000000000.0)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(-0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(0.25 * Float64(r * Float64(w * Float64(r * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (v <= 190000000000.0)
		tmp = -1.5 + ((2.0 / (r * r)) + (-0.375 * ((r * w) * (r * w))));
	else
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * (r * (w * (r * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[v, 190000000000.0], 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], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(0.25 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.9e11

   1. Initial program 87.6%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in v around 0 83.9%

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

      1. *-commutative 83.9%

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

      2. unpow2 83.9%

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

      3. unpow2 83.9%

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

      4. swap-sqr 97.5%

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

      5. unpow2 97.5%

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

   7. Simplified 97.5%

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

      1. unpow2 97.5%

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

   9. Applied egg-rr 97.5%

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

   if 1.9e11 < v

   1. Initial program 71.8%

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

   3. Simplified 96.4%

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

      1. div-inv 96.3%

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

      2. associate-*r* 99.7%

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

      3. associate-*l* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf 65.9%

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

      1. unpow2 65.9%

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

      2. unpow2 65.9%

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

      3. swap-sqr 99.7%

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

      4. unpow2 99.7%

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

   9. Simplified 99.7%

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

       1. unpow2 99.7%

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

       2. *-commutative 99.7%

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

       3. associate-*r* 96.3%

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

   11. Applied egg-rr 96.3%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 190000000000:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% 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 86.0%

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

5. Taylor expanded in v around 0 79.3%

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

   1. *-commutative 79.3%

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

   2. unpow2 79.3%

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

   3. unpow2 79.3%

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

   4. swap-sqr 94.4%

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

   5. unpow2 94.4%

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

7. Simplified 94.4%

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

   1. unpow2 94.4%

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

9. Applied egg-rr 94.4%

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

10. Final simplification 94.4%

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

Reproduce

herbie shell --seed 2024011 
(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))