Rosa's TurbineBenchmark

Percentage Accurate: 84.6% → 99.8%

Time: 14.7s

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: 84.6% 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, 1.0× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (+ 3.0 (/ 2.0 (* r r)))
  (- (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (/ (* r w) (- v 1.0)))) 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))) * ((r * w) * ((r * w) / (v - 1.0)))) - 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))) * ((r * w) * ((r * w) / (v - 1.0d0)))) - 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))) * ((r * w) * ((r * w) / (v - 1.0)))) - 4.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

   1. associate--l- 84.7%

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

   2. associate-*l* 79.1%

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

   3. sqr-neg 79.1%

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

   4. associate-*l* 84.7%

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

   5. associate-/l* 86.8%

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

   6. fma-define 86.9%

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

3. Simplified 86.8%

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

   1. add-sqr-sqrt 86.8%

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

   2. *-un-lft-identity 86.8%

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

   3. times-frac 86.8%

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

   4. associate-*r* 80.5%

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

   5. sqrt-prod 80.5%

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

   6. sqrt-prod 41.4%

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

   7. add-sqr-sqrt 67.2%

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

   8. sqrt-prod 40.9%

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

   9. add-sqr-sqrt 73.3%

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

   10. associate-*r* 65.0%

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

   11. sqrt-prod 65.0%

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

   12. sqrt-prod 36.4%

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

   13. add-sqr-sqrt 70.7%

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

   14. sqrt-prod 58.1%

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

   15. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1020000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25 + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{\frac{1 - v}{r} \cdot \frac{-1}{w}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1020000.0)
   (+ (/ 2.0 (* r r)) (+ (* (* (* r w) (* r w)) -0.25) -1.5))
   (+
    3.0
    (-
     (/ (* (* r w) (+ 0.375 (* v -0.25))) (* (/ (- 1.0 v) r) (/ -1.0 w)))
     4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1020000.0) {
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5);
	} else {
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / (((1.0 - v) / r) * (-1.0 / w))) - 4.5);
	}
	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 (r <= 1020000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((((r * w) * (r * w)) * (-0.25d0)) + (-1.5d0))
    else
        tmp = 3.0d0 + ((((r * w) * (0.375d0 + (v * (-0.25d0)))) / (((1.0d0 - v) / r) * ((-1.0d0) / w))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1020000.0) {
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5);
	} else {
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / (((1.0 - v) / r) * (-1.0 / w))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1020000.0:
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5)
	else:
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / (((1.0 - v) / r) * (-1.0 / w))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1020000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25) + -1.5));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(Float64(r * w) * Float64(0.375 + Float64(v * -0.25))) / Float64(Float64(Float64(1.0 - v) / r) * Float64(-1.0 / w))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1020000.0)
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5);
	else
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / (((1.0 - v) / r) * (-1.0 / w))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1020000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision] * N[(-1.0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.02e6

   1. Initial program 82.7%

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

   3. Simplified 85.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. unpow2 79.0%

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

      3. unpow2 79.0%

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

      4. swap-sqr 96.0%

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

      5. unpow2 96.0%

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

   7. Simplified 96.0%

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

      1. unpow2 96.0%

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

   9. Applied egg-rr 96.0%

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

   if 1.02e6 < r

   1. Initial program 92.5%

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

      1. associate--l- 92.5%

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

      2. associate-*l* 74.7%

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

      3. sqr-neg 74.7%

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

      4. associate-*l* 92.5%

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

      5. associate-/l* 92.5%

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

      6. fma-define 92.5%

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

   3. Simplified 92.5%

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

      1. add-sqr-sqrt 92.4%

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

      2. *-un-lft-identity 92.4%

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

      3. times-frac 92.4%

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

      4. associate-*r* 74.6%

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

      5. sqrt-prod 74.6%

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

      6. sqrt-prod 92.4%

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

      7. add-sqr-sqrt 92.4%

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

      8. sqrt-prod 55.3%

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

      9. add-sqr-sqrt 65.1%

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

      10. associate-*r* 49.1%

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

      11. sqrt-prod 49.2%

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

      12. sqrt-prod 65.1%

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

      13. add-sqr-sqrt 65.2%

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

      14. sqrt-prod 62.6%

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

      15. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around inf 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-*r* 99.8%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

         \[\leadsto 3 - \left(\frac{\left(\color{blue}{0.375} + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \left(r \cdot w\right)}{\frac{1 - v}{r \cdot w}} + 4.5\right) \]

      7. associate-*r* 99.8%

         \[\leadsto 3 - \left(\frac{\left(0.375 + \color{blue}{\left(0.125 \cdot -2\right) \cdot v}\right) \cdot \left(r \cdot w\right)}{\frac{1 - v}{r \cdot w}} + 4.5\right) \]

      8. metadata-eval 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-/r* 99.8%

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

       2. div-inv 99.8%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1020000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25 + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{\frac{1 - v}{r} \cdot \frac{-1}{w}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25 + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v - 1}\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 3.4e-6)
   (+ (/ 2.0 (* r r)) (+ (* (* (* r w) (* r w)) -0.25) -1.5))
   (+
    3.0
    (-
     (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (/ (* r w) (- v 1.0))))
     4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.4e-6) {
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5);
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * ((r * w) / (v - 1.0)))) - 4.5);
	}
	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 (r <= 3.4d-6) then
        tmp = (2.0d0 / (r * r)) + ((((r * w) * (r * w)) * (-0.25d0)) + (-1.5d0))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * ((r * w) / (v - 1.0d0)))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.4e-6) {
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5);
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * ((r * w) / (v - 1.0)))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 3.4e-6:
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5)
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * ((r * w) / (v - 1.0)))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3.4e-6)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25) + -1.5));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(v - 1.0)))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 3.4e-6)
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5);
	else
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * ((r * w) / (v - 1.0)))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3.4e-6], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(v - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.40000000000000006e-6

   1. Initial program 82.9%

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

   3. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. unpow2 79.0%

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

      3. unpow2 79.0%

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

      4. swap-sqr 96.3%

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

      5. unpow2 96.3%

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

   7. Simplified 96.3%

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

      1. unpow2 96.3%

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

   9. Applied egg-rr 96.3%

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

   if 3.40000000000000006e-6 < r

   1. Initial program 91.2%

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

      1. associate--l- 91.2%

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

      2. associate-*l* 74.4%

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

      3. sqr-neg 74.4%

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

      4. associate-*l* 91.2%

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

      5. associate-/l* 92.9%

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

      6. fma-define 92.9%

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

   3. Simplified 92.9%

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

      1. add-sqr-sqrt 92.8%

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

      2. *-un-lft-identity 92.8%

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

      3. times-frac 92.8%

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

      4. associate-*r* 76.0%

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

      5. sqrt-prod 76.0%

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

      6. sqrt-prod 92.8%

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

      7. add-sqr-sqrt 92.8%

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

      8. sqrt-prod 54.1%

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

      9. add-sqr-sqrt 63.4%

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

      10. associate-*r* 48.3%

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

      11. sqrt-prod 48.3%

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

      12. sqrt-prod 63.4%

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

      13. add-sqr-sqrt 63.4%

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

      14. sqrt-prod 61.0%

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

      15. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around inf 99.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25 + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v - 1}\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 12000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25 + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{\frac{v + -1}{r \cdot w}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 12000000.0)
   (+ (/ 2.0 (* r r)) (+ (* (* (* r w) (* r w)) -0.25) -1.5))
   (+
    3.0
    (- (/ (* (* r w) (+ 0.375 (* v -0.25))) (/ (+ v -1.0) (* r w))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 12000000.0) {
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5);
	} else {
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5);
	}
	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 (r <= 12000000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((((r * w) * (r * w)) * (-0.25d0)) + (-1.5d0))
    else
        tmp = 3.0d0 + ((((r * w) * (0.375d0 + (v * (-0.25d0)))) / ((v + (-1.0d0)) / (r * w))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 12000000.0) {
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5);
	} else {
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 12000000.0:
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5)
	else:
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 12000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25) + -1.5));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(Float64(r * w) * Float64(0.375 + Float64(v * -0.25))) / Float64(Float64(v + -1.0) / Float64(r * w))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 12000000.0)
		tmp = (2.0 / (r * r)) + ((((r * w) * (r * w)) * -0.25) + -1.5);
	else
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 12000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.2e7

   1. Initial program 82.7%

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

   3. Simplified 85.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. unpow2 79.0%

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

      3. unpow2 79.0%

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

      4. swap-sqr 96.0%

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

      5. unpow2 96.0%

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

   7. Simplified 96.0%

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

      1. unpow2 96.0%

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

   9. Applied egg-rr 96.0%

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

   if 1.2e7 < r

   1. Initial program 92.5%

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

      1. associate--l- 92.5%

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

      2. associate-*l* 74.7%

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

      3. sqr-neg 74.7%

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

      4. associate-*l* 92.5%

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

      5. associate-/l* 92.5%

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

      6. fma-define 92.5%

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

   3. Simplified 92.5%

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

      1. add-sqr-sqrt 92.4%

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

      2. *-un-lft-identity 92.4%

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

      3. times-frac 92.4%

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

      4. associate-*r* 74.6%

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

      5. sqrt-prod 74.6%

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

      6. sqrt-prod 92.4%

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

      7. add-sqr-sqrt 92.4%

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

      8. sqrt-prod 55.3%

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

      9. add-sqr-sqrt 65.1%

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

      10. associate-*r* 49.1%

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

      11. sqrt-prod 49.2%

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

      12. sqrt-prod 65.1%

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

      13. add-sqr-sqrt 65.2%

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

      14. sqrt-prod 62.6%

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

      15. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around inf 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. associate-*r* 99.8%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

         \[\leadsto 3 - \left(\frac{\left(\color{blue}{0.375} + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \left(r \cdot w\right)}{\frac{1 - v}{r \cdot w}} + 4.5\right) \]

      7. associate-*r* 99.8%

         \[\leadsto 3 - \left(\frac{\left(0.375 + \color{blue}{\left(0.125 \cdot -2\right) \cdot v}\right) \cdot \left(r \cdot w\right)}{\frac{1 - v}{r \cdot w}} + 4.5\right) \]

      8. metadata-eval 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 12000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25 + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{\frac{v + -1}{r \cdot w}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

3. Simplified 86.9%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 77.2%

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

   1. *-commutative 77.2%

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

   2. unpow2 77.2%

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

   3. unpow2 77.2%

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

   4. swap-sqr 94.3%

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

   5. unpow2 94.3%

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

7. Simplified 94.3%

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

   1. unpow2 94.3%

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

9. Applied egg-rr 94.3%

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

10. Final simplification 94.3%

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

Alternative 6: 57.4% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

3. Simplified 86.9%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 77.2%

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

   1. *-commutative 77.2%

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

   2. unpow2 77.2%

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

   3. unpow2 77.2%

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

   4. swap-sqr 94.3%

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

   5. unpow2 94.3%

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

7. Simplified 94.3%

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

8. Taylor expanded in r around 0 59.6%

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

9. Final simplification 59.6%

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

Alternative 7: 13.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

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

Fortran

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

Java

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

Python

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

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

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

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 84.7%

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

3. Simplified 80.3%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 59.6%

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

6. Taylor expanded in r around inf 16.4%

   \[\leadsto \color{blue}{3} - 4.5 \]

7. Final simplification 16.4%

   \[\leadsto -1.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(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))