Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.8%

Time: 10.9s

Alternatives: 8

Speedup: 1.1×

• 52.9% of points produce a very large (infinite) output. You may want to add a precondition.

• Rival profile vector overflowed, profile may not be complete

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: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (3.0 + (pow(r, -2.0) * 2.0)) + ((((v * -0.25) + 0.375) / ((v + -1.0) / ((r * w) * (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 + ((r ** (-2.0d0)) * 2.0d0)) + ((((v * (-0.25d0)) + 0.375d0) / ((v + (-1.0d0)) / ((r * w) * (r * w)))) - 4.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (3.0 + ((r ^ -2.0) * 2.0)) + ((((v * -0.25) + 0.375) / ((v + -1.0) / ((r * w) * (r * w)))) - 4.5);
end

Wolfram

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

Derivation

1. Initial program 86.5%

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

   1. associate--l- 86.5%

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

   2. associate-*l* 83.4%

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

   3. sqr-neg 83.4%

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

   4. associate-*l* 86.5%

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

   5. associate-/l* 87.6%

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

   6. fma-define 87.6%

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

3. Simplified 87.6%

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

   1. clear-num 87.6%

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

   2. un-div-inv 87.6%

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

   3. +-commutative 87.6%

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

   4. distribute-rgt-in 87.6%

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

   5. metadata-eval 87.6%

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

   6. *-commutative 87.6%

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

   7. associate-*l* 87.6%

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

   8. metadata-eval 87.6%

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

   9. associate-*r* 84.5%

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

   10. pow2 84.5%

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

   11. pow2 84.5%

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

   12. pow-prod-down 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

   1. clear-num 99.8%

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

   2. associate-/r/ 99.8%

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

   3. pow2 99.8%

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

   4. pow-flip 99.9%

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

   5. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (3.0 + (2.0 / (r * r))) + ((((v * -0.25) + 0.375) / (pow((r * w), -2.0) * (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))) + ((((v * (-0.25d0)) + 0.375d0) / (((r * w) ** (-2.0d0)) * (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))) + ((((v * -0.25) + 0.375) / (Math.pow((r * w), -2.0) * (v + -1.0))) - 4.5);
}

Python

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

MATLAB

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

Derivation

1. Initial program 86.5%

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

   1. associate--l- 86.5%

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

   2. associate-*l* 83.4%

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

   3. sqr-neg 83.4%

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

   4. associate-*l* 86.5%

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

   5. associate-/l* 87.6%

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

   6. fma-define 87.6%

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

3. Simplified 87.6%

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

   1. clear-num 87.6%

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

   2. un-div-inv 87.6%

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

   3. +-commutative 87.6%

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

   4. distribute-rgt-in 87.6%

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

   5. metadata-eval 87.6%

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

   6. *-commutative 87.6%

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

   7. associate-*l* 87.6%

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

   8. metadata-eval 87.6%

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

   9. associate-*r* 84.5%

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

   10. pow2 84.5%

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

   11. pow2 84.5%

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

   12. pow-prod-down 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. div-inv 99.8%

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

   3. pow-flip 99.8%

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

   4. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{if}\;r \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(4.5 + t\_0 \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{t\_0}} + 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))))
   (if (<= r 5.7e-5)
     (- (+ 3.0 (/ (/ 2.0 r) r)) (+ 4.5 (* t_0 (* 0.125 (+ 3.0 (* -2.0 v))))))
     (- 3.0 (+ (/ (+ (* v -0.25) 0.375) (/ (- 1.0 v) t_0)) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if (r <= 5.7e-5) {
		tmp = (3.0 + ((2.0 / r) / r)) - (4.5 + (t_0 * (0.125 * (3.0 + (-2.0 * v)))));
	} else {
		tmp = 3.0 - ((((v * -0.25) + 0.375) / ((1.0 - v) / t_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) :: t_0
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    if (r <= 5.7d-5) then
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - (4.5d0 + (t_0 * (0.125d0 * (3.0d0 + ((-2.0d0) * v)))))
    else
        tmp = 3.0d0 - ((((v * (-0.25d0)) + 0.375d0) / ((1.0d0 - v) / t_0)) + 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if (r <= 5.7e-5) {
		tmp = (3.0 + ((2.0 / r) / r)) - (4.5 + (t_0 * (0.125 * (3.0 + (-2.0 * v)))));
	} else {
		tmp = 3.0 - ((((v * -0.25) + 0.375) / ((1.0 - v) / t_0)) + 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	tmp = 0
	if r <= 5.7e-5:
		tmp = (3.0 + ((2.0 / r) / r)) - (4.5 + (t_0 * (0.125 * (3.0 + (-2.0 * v)))))
	else:
		tmp = 3.0 - ((((v * -0.25) + 0.375) / ((1.0 - v) / t_0)) + 4.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	tmp = 0.0
	if (r <= 5.7e-5)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - Float64(4.5 + Float64(t_0 * Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))))));
	else
		tmp = Float64(3.0 - Float64(Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(Float64(1.0 - v) / t_0)) + 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	tmp = 0.0;
	if (r <= 5.7e-5)
		tmp = (3.0 + ((2.0 / r) / r)) - (4.5 + (t_0 * (0.125 * (3.0 + (-2.0 * v)))));
	else
		tmp = 3.0 - ((((v * -0.25) + 0.375) / ((1.0 - v) / t_0)) + 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 5.7e-5], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - N[(4.5 + N[(t$95$0 * N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 5.7000000000000003e-5

   1. Initial program 85.0%

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

      1. associate--l- 85.0%

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

      2. associate-*l* 83.6%

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

      3. sqr-neg 83.6%

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

      4. associate-*l* 85.0%

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

      5. associate-/l* 85.5%

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

      6. fma-define 85.5%

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

   3. Simplified 85.5%

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

      1. associate-/r* 85.5%

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

      2. div-inv 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-*r/ 85.5%

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

      2. *-rgt-identity 85.5%

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

   8. Simplified 85.5%

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

      1. div-inv 85.5%

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

      2. add-sqr-sqrt 85.5%

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

      3. associate-*l* 85.5%

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

      4. sqrt-prod 29.1%

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

      5. sqrt-prod 29.1%

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

      6. sqrt-prod 14.0%

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

      7. add-sqr-sqrt 29.1%

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

      8. associate-*l* 29.1%

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

      9. add-sqr-sqrt 71.5%

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

      10. sqrt-prod 29.1%

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

      11. sqrt-prod 29.1%

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

      12. sqrt-prod 16.0%

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

      13. add-sqr-sqrt 34.1%

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

      14. associate-*l* 34.1%

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

      15. add-sqr-sqrt 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in v around 0 85.3%

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

   if 5.7000000000000003e-5 < r

   1. Initial program 91.0%

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

      1. associate--l- 91.0%

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

      2. associate-*l* 82.7%

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

      3. sqr-neg 82.7%

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

      4. associate-*l* 91.0%

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

      5. associate-/l* 94.0%

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

      6. fma-define 94.0%

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

   3. Simplified 94.0%

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

      1. clear-num 94.0%

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

      2. un-div-inv 94.0%

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

      3. +-commutative 94.0%

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

      4. distribute-rgt-in 94.0%

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

      5. metadata-eval 94.0%

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

      6. *-commutative 94.0%

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

      7. associate-*l* 94.0%

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

      8. metadata-eval 94.0%

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

      9. associate-*r* 85.7%

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

      10. pow2 85.7%

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

      11. pow2 85.7%

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

      12. pow-prod-down 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in r around inf 99.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(4.5 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}} + 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (3.0 + (2.0 / (r * r))) + ((((v * -0.25) + 0.375) / ((v + -1.0) / ((r * w) * (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))) + ((((v * (-0.25d0)) + 0.375d0) / ((v + (-1.0d0)) / ((r * w) * (r * w)))) - 4.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (3.0 + (2.0 / (r * r))) + ((((v * -0.25) + 0.375) / ((v + -1.0) / ((r * w) * (r * w)))) - 4.5);
end

Wolfram

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

Derivation

1. Initial program 86.5%

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

   1. associate--l- 86.5%

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

   2. associate-*l* 83.4%

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

   3. sqr-neg 83.4%

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

   4. associate-*l* 86.5%

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

   5. associate-/l* 87.6%

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

   6. fma-define 87.6%

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

3. Simplified 87.6%

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

   1. clear-num 87.6%

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

   2. un-div-inv 87.6%

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

   3. +-commutative 87.6%

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

   4. distribute-rgt-in 87.6%

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

   5. metadata-eval 87.6%

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

   6. *-commutative 87.6%

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

   7. associate-*l* 87.6%

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

   8. metadata-eval 87.6%

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

   9. associate-*r* 84.5%

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

   10. pow2 84.5%

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

   11. pow2 84.5%

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

   12. pow-prod-down 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}} + 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.6e-5)
   (- (+ 3.0 (/ 2.0 (* r r))) 4.5)
   (-
    3.0
    (+ (/ (+ (* v -0.25) 0.375) (/ (- 1.0 v) (* (* r w) (* r w)))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.6e-5) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else {
		tmp = 3.0 - ((((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (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 <= 5.6d-5) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - 4.5d0
    else
        tmp = 3.0d0 - ((((v * (-0.25d0)) + 0.375d0) / ((1.0d0 - v) / ((r * w) * (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 <= 5.6e-5) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else {
		tmp = 3.0 - ((((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w)))) + 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.6e-5:
		tmp = (3.0 + (2.0 / (r * r))) - 4.5
	else:
		tmp = 3.0 - ((((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w)))) + 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.6e-5)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - 4.5);
	else
		tmp = Float64(3.0 - Float64(Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(Float64(1.0 - v) / Float64(Float64(r * w) * Float64(r * w)))) + 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.6e-5)
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	else
		tmp = 3.0 - ((((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w)))) + 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.6e-5], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 - N[(N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.59999999999999992e-5

   1. Initial program 85.0%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in r around 0 70.0%

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

   if 5.59999999999999992e-5 < r

   1. Initial program 91.0%

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

      1. associate--l- 91.0%

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

      2. associate-*l* 82.7%

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

      3. sqr-neg 82.7%

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

      4. associate-*l* 91.0%

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

      5. associate-/l* 94.0%

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

      6. fma-define 94.0%

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

   3. Simplified 94.0%

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

      1. clear-num 94.0%

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

      2. un-div-inv 94.0%

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

      3. +-commutative 94.0%

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

      4. distribute-rgt-in 94.0%

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

      5. metadata-eval 94.0%

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

      6. *-commutative 94.0%

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

      7. associate-*l* 94.0%

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

      8. metadata-eval 94.0%

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

      9. associate-*r* 85.7%

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

      10. pow2 85.7%

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

      11. pow2 85.7%

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

      12. pow-prod-down 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in r around inf 99.8%

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

Alternative 6: 67.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.4e-5)
   (- (+ 3.0 (/ 2.0 (* r r))) 4.5)
   (- 3.0 (+ 4.5 (* (* (* r w) (* r w)) (* 0.125 (+ 3.0 (* -2.0 v))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.4e-5) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (-2.0 * v)))));
	}
	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 <= 5.4d-5) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - 4.5d0
    else
        tmp = 3.0d0 - (4.5d0 + (((r * w) * (r * w)) * (0.125d0 * (3.0d0 + ((-2.0d0) * v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.4e-5) {
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (-2.0 * v)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.4e-5:
		tmp = (3.0 + (2.0 / (r * r))) - 4.5
	else:
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (-2.0 * v)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.4e-5)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - 4.5);
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * Float64(r * w)) * Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.4e-5)
		tmp = (3.0 + (2.0 / (r * r))) - 4.5;
	else
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (-2.0 * v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.4e-5], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.3999999999999998e-5

   1. Initial program 85.0%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in r around 0 70.0%

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

   if 5.3999999999999998e-5 < r

   1. Initial program 91.0%

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

      1. associate--l- 91.0%

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

      2. associate-*l* 82.7%

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

      3. sqr-neg 82.7%

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

      4. associate-*l* 91.0%

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

      5. associate-/l* 94.0%

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

      6. fma-define 94.0%

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

   3. Simplified 94.0%

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

      1. associate-/r* 94.0%

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

      2. div-inv 94.0%

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

   6. Applied egg-rr 94.0%

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

      1. associate-*r/ 94.0%

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

      2. *-rgt-identity 94.0%

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

   8. Simplified 94.0%

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

      1. div-inv 93.9%

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

      2. add-sqr-sqrt 93.9%

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

      3. associate-*l* 93.9%

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

      4. sqrt-prod 93.9%

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

      5. sqrt-prod 93.9%

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

      6. sqrt-prod 49.3%

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

      7. add-sqr-sqrt 65.4%

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

      8. associate-*l* 65.4%

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

      9. add-sqr-sqrt 65.5%

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

      10. sqrt-prod 65.5%

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

      11. sqrt-prod 65.4%

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

      12. sqrt-prod 52.3%

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

      13. add-sqr-sqrt 99.8%

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

      14. associate-*l* 99.8%

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

      15. add-sqr-sqrt 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in r around inf 99.8%

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

   12. Taylor expanded in v around 0 64.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

3. Simplified 83.8%

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

5. Taylor expanded in r around 0 60.9%

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

Alternative 8: 13.7% 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 86.5%

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

3. Simplified 83.8%

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

5. Taylor expanded in r around 0 60.9%

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

6. Taylor expanded in r around inf 14.5%

   \[\leadsto \color{blue}{-1.5} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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))