Rosa's TurbineBenchmark

Percentage Accurate: 84.1% → 99.7%

Time: 13.4s

Alternatives: 10

Speedup: 1.5×

• 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: 84.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Simplified 87.9%

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

   1. div-inv 87.9%

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

   2. distribute-lft-in 87.9%

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

   3. metadata-eval 87.9%

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

   4. associate-*r* 87.9%

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

   5. metadata-eval 87.9%

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

   6. associate-*r* 96.8%

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

   7. *-commutative 96.8%

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

   8. clear-num 96.9%

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

   9. associate-*r* 99.8%

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

   10. pow2 99.8%

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

   11. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -41000000 \lor \neg \left(v \leq 9.8 \cdot 10^{-36}\right):\\ \;\;\;\;t_0 - \left(4.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(4.5 + \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (or (<= v -41000000.0) (not (<= v 9.8e-36)))
     (- t_0 (+ 4.5 (* (* r w) (* (* r w) 0.25))))
     (- t_0 (+ 4.5 (* (* r w) (* 0.375 (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if ((v <= -41000000.0) || !(v <= 9.8e-36)) {
		tmp = t_0 - (4.5 + ((r * w) * ((r * w) * 0.25)));
	} else {
		tmp = t_0 - (4.5 + ((r * w) * (0.375 * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    if ((v <= (-41000000.0d0)) .or. (.not. (v <= 9.8d-36))) then
        tmp = t_0 - (4.5d0 + ((r * w) * ((r * w) * 0.25d0)))
    else
        tmp = t_0 - (4.5d0 + ((r * w) * (0.375d0 * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if ((v <= -41000000.0) || !(v <= 9.8e-36)) {
		tmp = t_0 - (4.5 + ((r * w) * ((r * w) * 0.25)));
	} else {
		tmp = t_0 - (4.5 + ((r * w) * (0.375 * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if (v <= -41000000.0) or not (v <= 9.8e-36):
		tmp = t_0 - (4.5 + ((r * w) * ((r * w) * 0.25)))
	else:
		tmp = t_0 - (4.5 + ((r * w) * (0.375 * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if ((v <= -41000000.0) || !(v <= 9.8e-36))
		tmp = Float64(t_0 - Float64(4.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * 0.25))));
	else
		tmp = Float64(t_0 - Float64(4.5 + Float64(Float64(r * w) * Float64(0.375 * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if ((v <= -41000000.0) || ~((v <= 9.8e-36)))
		tmp = t_0 - (4.5 + ((r * w) * ((r * w) * 0.25)));
	else
		tmp = t_0 - (4.5 + ((r * w) * (0.375 * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -41000000.0], N[Not[LessEqual[v, 9.8e-36]], $MachinePrecision]], N[(t$95$0 - N[(4.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(4.5 + N[(N[(r * w), $MachinePrecision] * N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -4.1e7 or 9.7999999999999994e-36 < v

   1. Initial program 79.5%

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

   3. Simplified 86.4%

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

      1. associate-/r/ 86.4%

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

      2. associate-*r* 83.7%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf 99.3%

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

   if -4.1e7 < v < 9.7999999999999994e-36

   1. Initial program 89.8%

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

   3. Simplified 89.8%

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

      1. associate-/r/ 89.8%

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

      2. associate-*r* 84.4%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -41000000 \lor \neg \left(v \leq 9.8 \cdot 10^{-36}\right):\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -41000000 \lor \neg \left(v \leq 9.8 \cdot 10^{-36}\right):\\ \;\;\;\;t_0 - \left(4.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(4.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (or (<= v -41000000.0) (not (<= v 9.8e-36)))
     (- t_0 (+ 4.5 (* (* r w) (* (* r w) 0.25))))
     (- t_0 (+ 4.5 (* (* r w) (* w (* r 0.375))))))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if ((v <= -41000000.0) || !(v <= 9.8e-36)) {
		tmp = t_0 - (4.5 + ((r * w) * ((r * w) * 0.25)));
	} else {
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    if ((v <= (-41000000.0d0)) .or. (.not. (v <= 9.8d-36))) then
        tmp = t_0 - (4.5d0 + ((r * w) * ((r * w) * 0.25d0)))
    else
        tmp = t_0 - (4.5d0 + ((r * w) * (w * (r * 0.375d0))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if ((v <= -41000000.0) || !(v <= 9.8e-36)) {
		tmp = t_0 - (4.5 + ((r * w) * ((r * w) * 0.25)));
	} else {
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if (v <= -41000000.0) or not (v <= 9.8e-36):
		tmp = t_0 - (4.5 + ((r * w) * ((r * w) * 0.25)))
	else:
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if ((v <= -41000000.0) || !(v <= 9.8e-36))
		tmp = Float64(t_0 - Float64(4.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * 0.25))));
	else
		tmp = Float64(t_0 - Float64(4.5 + Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if ((v <= -41000000.0) || ~((v <= 9.8e-36)))
		tmp = t_0 - (4.5 + ((r * w) * ((r * w) * 0.25)));
	else
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -41000000.0], N[Not[LessEqual[v, 9.8e-36]], $MachinePrecision]], N[(t$95$0 - N[(4.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(4.5 + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -4.1e7 or 9.7999999999999994e-36 < v

   1. Initial program 79.5%

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

   3. Simplified 86.4%

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

      1. associate-/r/ 86.4%

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

      2. associate-*r* 83.7%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf 99.3%

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

   if -4.1e7 < v < 9.7999999999999994e-36

   1. Initial program 89.8%

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

   3. Simplified 89.8%

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

      1. associate-/r/ 89.8%

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

      2. associate-*r* 84.4%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 99.8%

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

      1. associate-*r* 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -41000000 \lor \neg \left(v \leq 9.8 \cdot 10^{-36}\right):\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := 4.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\\ \mathbf{if}\;v \leq -41000000:\\ \;\;\;\;t_0 - t_1\\ \mathbf{elif}\;v \leq 1.55 \cdot 10^{-8}:\\ \;\;\;\;t_0 - \left(4.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1 (+ 4.5 (* (* r w) (* (* r w) 0.25)))))
   (if (<= v -41000000.0)
     (- t_0 t_1)
     (if (<= v 1.55e-8)
       (- t_0 (+ 4.5 (* (* r w) (* w (* r 0.375)))))
       (- (+ 3.0 (/ (/ 2.0 r) r)) t_1)))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = 4.5 + ((r * w) * ((r * w) * 0.25));
	double tmp;
	if (v <= -41000000.0) {
		tmp = t_0 - t_1;
	} else if (v <= 1.55e-8) {
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))));
	} else {
		tmp = (3.0 + ((2.0 / r) / r)) - t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    t_1 = 4.5d0 + ((r * w) * ((r * w) * 0.25d0))
    if (v <= (-41000000.0d0)) then
        tmp = t_0 - t_1
    else if (v <= 1.55d-8) then
        tmp = t_0 - (4.5d0 + ((r * w) * (w * (r * 0.375d0))))
    else
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = 4.5 + ((r * w) * ((r * w) * 0.25));
	double tmp;
	if (v <= -41000000.0) {
		tmp = t_0 - t_1;
	} else if (v <= 1.55e-8) {
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))));
	} else {
		tmp = (3.0 + ((2.0 / r) / r)) - t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	t_1 = 4.5 + ((r * w) * ((r * w) * 0.25))
	tmp = 0
	if v <= -41000000.0:
		tmp = t_0 - t_1
	elif v <= 1.55e-8:
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))))
	else:
		tmp = (3.0 + ((2.0 / r) / r)) - t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(4.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * 0.25)))
	tmp = 0.0
	if (v <= -41000000.0)
		tmp = Float64(t_0 - t_1);
	elseif (v <= 1.55e-8)
		tmp = Float64(t_0 - Float64(4.5 + Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	t_1 = 4.5 + ((r * w) * ((r * w) * 0.25));
	tmp = 0.0;
	if (v <= -41000000.0)
		tmp = t_0 - t_1;
	elseif (v <= 1.55e-8)
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))));
	else
		tmp = (3.0 + ((2.0 / r) / r)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -41000000.0], N[(t$95$0 - t$95$1), $MachinePrecision], If[LessEqual[v, 1.55e-8], N[(t$95$0 - N[(4.5 + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.1e7

   1. Initial program 79.8%

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

   3. Simplified 85.1%

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

      1. associate-/r/ 85.1%

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

      2. associate-*r* 81.9%

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

      3. swap-sqr 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-*r* 99.9%

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

      8. metadata-eval 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around inf 99.5%

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

   if -4.1e7 < v < 1.55e-8

   1. Initial program 89.5%

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

   3. Simplified 89.5%

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

      1. associate-/r/ 89.5%

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

      2. associate-*r* 84.5%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 99.8%

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

      1. associate-*r* 99.8%

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

   9. Simplified 99.8%

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

   if 1.55e-8 < v

   1. Initial program 78.5%

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

   3. Simplified 87.9%

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

      1. associate-/r/ 87.9%

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

      2. associate-*r* 85.3%

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

      3. swap-sqr 99.6%

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

      4. associate-*r* 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. associate-*r* 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-/r* 46.0%

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

      2. div-inv 46.0%

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

      3. *-un-lft-identity 46.0%

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

      4. times-frac 46.0%

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

      5. metadata-eval 46.0%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*r/ 46.0%

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

      2. associate-*l/ 45.9%

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

      3. associate-*r/ 46.0%

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

      4. *-rgt-identity 46.0%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in v around inf 99.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -41000000:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\\ \mathbf{elif}\;v \leq 1.55 \cdot 10^{-8}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(4.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{\frac{2}{r}}{r}\\ t_1 := 4.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\\ \mathbf{if}\;v \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - t_1\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-37}:\\ \;\;\;\;t_0 - \left(4.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ (/ 2.0 r) r)))
        (t_1 (+ 4.5 (* (* r w) (* (* r w) 0.25)))))
   (if (<= v -3.1e-52)
     (- (+ 3.0 (/ 2.0 (* r r))) t_1)
     (if (<= v 4e-37)
       (- t_0 (+ 4.5 (* (* r w) (* w (* r 0.375)))))
       (- t_0 t_1)))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + ((2.0 / r) / r);
	double t_1 = 4.5 + ((r * w) * ((r * w) * 0.25));
	double tmp;
	if (v <= -3.1e-52) {
		tmp = (3.0 + (2.0 / (r * r))) - t_1;
	} else if (v <= 4e-37) {
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))));
	} else {
		tmp = t_0 - t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + ((2.0d0 / r) / r)
    t_1 = 4.5d0 + ((r * w) * ((r * w) * 0.25d0))
    if (v <= (-3.1d-52)) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - t_1
    else if (v <= 4d-37) then
        tmp = t_0 - (4.5d0 + ((r * w) * (w * (r * 0.375d0))))
    else
        tmp = t_0 - t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + ((2.0 / r) / r);
	double t_1 = 4.5 + ((r * w) * ((r * w) * 0.25));
	double tmp;
	if (v <= -3.1e-52) {
		tmp = (3.0 + (2.0 / (r * r))) - t_1;
	} else if (v <= 4e-37) {
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))));
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + ((2.0 / r) / r)
	t_1 = 4.5 + ((r * w) * ((r * w) * 0.25))
	tmp = 0
	if v <= -3.1e-52:
		tmp = (3.0 + (2.0 / (r * r))) - t_1
	elif v <= 4e-37:
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))))
	else:
		tmp = t_0 - t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(Float64(2.0 / r) / r))
	t_1 = Float64(4.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * 0.25)))
	tmp = 0.0
	if (v <= -3.1e-52)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - t_1);
	elseif (v <= 4e-37)
		tmp = Float64(t_0 - Float64(4.5 + Float64(Float64(r * w) * Float64(w * Float64(r * 0.375)))));
	else
		tmp = Float64(t_0 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + ((2.0 / r) / r);
	t_1 = 4.5 + ((r * w) * ((r * w) * 0.25));
	tmp = 0.0;
	if (v <= -3.1e-52)
		tmp = (3.0 + (2.0 / (r * r))) - t_1;
	elseif (v <= 4e-37)
		tmp = t_0 - (4.5 + ((r * w) * (w * (r * 0.375))));
	else
		tmp = t_0 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -3.1e-52], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[v, 4e-37], N[(t$95$0 - N[(4.5 + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.0999999999999999e-52

   1. Initial program 81.1%

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

   3. Simplified 86.1%

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

      1. associate-/r/ 86.1%

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

      2. associate-*r* 83.1%

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

      3. swap-sqr 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-*r* 99.9%

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

      8. metadata-eval 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around inf 99.5%

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

   if -3.0999999999999999e-52 < v < 4.00000000000000027e-37

   1. Initial program 89.3%

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

   3. Simplified 89.3%

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

      1. associate-/r/ 89.3%

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

      2. associate-*r* 83.7%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-/r* 56.6%

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

      2. div-inv 56.6%

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

      3. *-un-lft-identity 56.6%

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

      4. times-frac 56.6%

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

      5. metadata-eval 56.6%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*r/ 56.6%

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

      2. associate-*l/ 56.5%

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

      3. associate-*r/ 56.6%

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

      4. *-rgt-identity 56.6%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in v around 0 99.8%

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

       1. associate-*r* 99.8%

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

   13. Simplified 99.8%

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

   if 4.00000000000000027e-37 < v

   1. Initial program 79.2%

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

   3. Simplified 87.7%

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

      1. associate-/r/ 87.7%

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

      2. associate-*r* 85.4%

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

      3. swap-sqr 99.7%

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

      4. associate-*r* 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. metadata-eval 99.7%

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

      7. associate-*r* 99.7%

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

      8. metadata-eval 99.7%

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

      9. *-commutative 99.7%

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

      10. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-/r* 49.8%

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

      2. div-inv 49.8%

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

      3. *-un-lft-identity 49.8%

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

      4. times-frac 49.8%

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

      5. metadata-eval 49.8%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*r/ 49.8%

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

      2. associate-*l/ 49.7%

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

      3. associate-*r/ 49.8%

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

      4. *-rgt-identity 49.8%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in v around inf 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-37}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(4.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(4.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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 + ((r * w) * ((r * w) * ((0.375d0 + ((-0.25d0) * v)) / (1.0d0 - v)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Simplified 87.9%

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

   1. associate-/r/ 87.9%

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

   2. associate-*r* 84.0%

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

   3. swap-sqr 99.8%

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

   4. associate-*r* 99.8%

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

   5. distribute-lft-in 99.8%

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

   6. metadata-eval 99.8%

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

   7. associate-*r* 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 7: 93.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 + ((r * w) * ((r * w) * 0.25d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Simplified 87.9%

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

   1. associate-/r/ 87.9%

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

   2. associate-*r* 84.0%

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

   3. swap-sqr 99.8%

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

   4. associate-*r* 99.8%

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

   5. distribute-lft-in 99.8%

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

   6. metadata-eval 99.8%

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

   7. associate-*r* 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in v around inf 93.8%

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

8. Final simplification 93.8%

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

Alternative 8: 56.7% 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 84.0%

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

3. Simplified 87.9%

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

   1. associate-/r/ 87.9%

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

   2. associate-*r* 84.0%

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

   3. swap-sqr 99.8%

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

   4. associate-*r* 99.8%

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

   5. distribute-lft-in 99.8%

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

   6. metadata-eval 99.8%

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

   7. associate-*r* 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in w around 0 54.1%

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

8. Final simplification 54.1%

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

Alternative 9: 56.7% accurate, 3.2× speedup

Math

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

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(Float64(2.0 / 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[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 84.0%

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

3. Simplified 87.9%

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

   1. associate-/r/ 87.9%

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

   2. associate-*r* 84.0%

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

   3. swap-sqr 99.8%

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

   4. associate-*r* 99.8%

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

   5. distribute-lft-in 99.8%

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

   6. metadata-eval 99.8%

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

   7. associate-*r* 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in w around 0 54.1%

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

   1. associate-/r* 54.1%

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

   2. div-inv 54.1%

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

   3. *-un-lft-identity 54.1%

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

   4. times-frac 54.1%

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

   5. metadata-eval 54.1%

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

9. Applied egg-rr 54.1%

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

    1. associate-*r/ 54.1%

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

    2. associate-*l/ 54.0%

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

    3. associate-*r/ 54.1%

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

    4. *-rgt-identity 54.1%

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

11. Simplified 54.1%

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

12. Final simplification 54.1%

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

Alternative 10: 14.3% 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 84.0%

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

3. Simplified 87.9%

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

   1. associate-/r/ 87.9%

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

   2. associate-*r* 84.0%

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

   3. swap-sqr 99.8%

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

   4. associate-*r* 99.8%

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

   5. distribute-lft-in 99.8%

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

   6. metadata-eval 99.8%

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

   7. associate-*r* 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in w around 0 54.1%

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

8. Taylor expanded in r around inf 12.3%

   \[\leadsto \color{blue}{-1.5} \]

9. Final simplification 12.3%

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

Reproduce

herbie shell --seed 2024018 
(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))