Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 99.8%

Time: 11.4s

Alternatives: 9

Speedup: 1.2×

• 52.6% 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: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (2.0d0 / (r * r)) + ((-1.5d0) + ((0.375d0 + ((-0.25d0) * v)) / (((v + (-1.0d0)) / (r * w)) / (r * w))))
end function

Java

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

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (-0.25 * v)) / (((v + -1.0) / (r * w)) / (r * w))))

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + ((0.375 + (-0.25 * v)) / (((v + -1.0) / (r * w)) / (r * w))));
end

Wolfram

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

Derivation

1. Initial program 84.9%

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

3. Simplified 89.3%

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

   1. associate-*r/ 90.1%

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

   2. *-commutative 90.1%

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

   3. associate-/l* 90.3%

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

   4. clear-num 90.3%

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

   5. un-div-inv 90.3%

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

   6. fma-undefine 90.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   7. *-commutative 90.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   8. distribute-lft-in 90.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.125 \cdot \left(-2 \cdot v\right) + 0.125 \cdot 3}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   9. metadata-eval 90.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \left(-2 \cdot v\right) + \color{blue}{0.375}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   10. associate-*r* 90.3%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(0.125 \cdot -2\right) \cdot v} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. metadata-eval 90.3%

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

   12. associate-*r* 83.7%

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

   13. pow2 83.7%

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

   14. pow2 83.7%

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

   15. pow-prod-down 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. times-frac 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.8%

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

    2. *-un-lft-identity 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -175 \lor \neg \left(v \leq 1.5\right):\\ \;\;\;\;t\_0 + \left(-1.5 + \left(-0.25 \cdot v\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -175.0) (not (<= v 1.5)))
     (+ t_0 (+ -1.5 (* (* -0.25 v) (/ (* r (* r (* w w))) (+ v -1.0)))))
     (+ t_0 (- -1.5 (* 0.375 (* (* r w) (/ (* r w) (- 1.0 v)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -175.0) || !(v <= 1.5)) {
		tmp = t_0 + (-1.5 + ((-0.25 * v) * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-175.0d0)) .or. (.not. (v <= 1.5d0))) then
        tmp = t_0 + ((-1.5d0) + (((-0.25d0) * v) * ((r * (r * (w * w))) / (v + (-1.0d0)))))
    else
        tmp = t_0 + ((-1.5d0) - (0.375d0 * ((r * w) * ((r * w) / (1.0d0 - v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -175.0) || !(v <= 1.5)) {
		tmp = t_0 + (-1.5 + ((-0.25 * v) * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -175.0) or not (v <= 1.5):
		tmp = t_0 + (-1.5 + ((-0.25 * v) * ((r * (r * (w * w))) / (v + -1.0))))
	else:
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -175.0) || !(v <= 1.5))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(-0.25 * v) * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(v + -1.0)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -175.0) || ~((v <= 1.5)))
		tmp = t_0 + (-1.5 + ((-0.25 * v) * ((r * (r * (w * w))) / (v + -1.0))));
	else
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -175.0], N[Not[LessEqual[v, 1.5]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(-0.25 * v), $MachinePrecision] * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -175 or 1.5 < v

   1. Initial program 79.9%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in v around inf 89.9%

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

      1. *-commutative 89.9%

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

   7. Simplified 89.9%

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

   if -175 < v < 1.5

   1. Initial program 90.2%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in v around 0 90.0%

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

      1. add-sqr-sqrt 90.0%

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

      2. *-un-lft-identity 90.0%

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

      3. times-frac 90.0%

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

      4. associate-*r* 85.1%

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

      5. sqrt-prod 85.1%

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

      6. sqrt-prod 48.7%

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

      7. add-sqr-sqrt 79.7%

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

      8. sqrt-prod 46.4%

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

      9. add-sqr-sqrt 79.4%

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

      10. associate-*r* 71.3%

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

      11. sqrt-prod 71.3%

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

      12. sqrt-prod 42.7%

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

      13. add-sqr-sqrt 78.3%

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

      14. sqrt-prod 55.8%

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

      15. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(0.375 \cdot \color{blue}{\left(\frac{r \cdot w}{1} \cdot \frac{r \cdot w}{v + -1}\right)} + -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -175 \lor \neg \left(v \leq 1.5\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(-0.25 \cdot v\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3400:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 3400.0)
   (+ (/ 2.0 (* r r)) (- -1.5 (* 0.375 (* (* r w) (/ (* r w) (- 1.0 v))))))
   (+
    3.0
    (-
     (* (* 0.125 (+ 3.0 (* v -2.0))) (* (* r w) (/ (* r w) (+ v -1.0))))
     4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3400.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / (v + -1.0)))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 3400.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (0.375d0 * ((r * w) * ((r * w) / (1.0d0 - v)))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * ((r * w) * ((r * w) / (v + (-1.0d0))))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 3400.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / (v + -1.0)))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 3400.0:
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / (v + -1.0)))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3400.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(v + -1.0)))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 3400.0)
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / (v + -1.0)))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3400.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3400

   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 89.3%

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

   5. Taylor expanded in v around 0 77.7%

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

      1. add-sqr-sqrt 77.7%

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

      2. *-un-lft-identity 77.7%

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

      3. times-frac 77.7%

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

      4. associate-*r* 75.1%

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

      5. sqrt-prod 75.1%

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

      6. sqrt-prod 32.8%

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

      7. add-sqr-sqrt 71.2%

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

      8. sqrt-prod 43.2%

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

      9. add-sqr-sqrt 76.2%

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

      10. associate-*r* 71.0%

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

      11. sqrt-prod 69.5%

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

      12. sqrt-prod 33.3%

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

      13. add-sqr-sqrt 73.6%

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

      14. sqrt-prod 48.7%

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

      15. add-sqr-sqrt 85.3%

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

   7. Applied egg-rr 85.3%

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

   if 3400 < r

   1. Initial program 87.6%

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

      1. associate--l- 87.6%

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

      2. associate-*l* 71.1%

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

      3. sqr-neg 71.1%

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

      4. associate-*l* 87.6%

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

      5. associate-/l* 93.7%

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

      6. fma-define 93.7%

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

   3. Simplified 93.7%

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

      1. add-sqr-sqrt 93.7%

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

      2. *-un-lft-identity 93.7%

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

      3. times-frac 93.6%

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

      4. associate-*r* 77.2%

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

      5. sqrt-prod 77.2%

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

      6. sqrt-prod 93.6%

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

      7. add-sqr-sqrt 93.7%

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

      8. sqrt-prod 50.9%

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

      9. add-sqr-sqrt 65.2%

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

      10. associate-*r* 50.2%

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

      11. sqrt-prod 50.2%

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

      12. sqrt-prod 65.1%

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

      13. add-sqr-sqrt 65.2%

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

      14. sqrt-prod 53.9%

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

      15. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. *-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3400:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.82 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= v 1.82e+21)
   (+ (/ 2.0 (* r r)) (- -1.5 (* 0.375 (* (* r w) (/ (* r w) (- 1.0 v))))))
   (+ 3.0 (- (* (* 0.125 (+ 3.0 (* v -2.0))) (* (* r w) (/ (* r w) v))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (v <= 1.82e+21) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (v <= 1.82d+21) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (0.375d0 * ((r * w) * ((r * w) / (1.0d0 - v)))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * ((r * w) * ((r * w) / v))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (v <= 1.82e+21) {
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if v <= 1.82e+21:
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (v <= 1.82e+21)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(Float64(r * w) * Float64(Float64(r * w) / v))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (v <= 1.82e+21)
		tmp = (2.0 / (r * r)) + (-1.5 - (0.375 * ((r * w) * ((r * w) / (1.0 - v)))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[v, 1.82e+21], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.82e21

   1. Initial program 87.5%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in v around 0 85.4%

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

      1. add-sqr-sqrt 85.4%

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

      2. *-un-lft-identity 85.4%

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

      3. times-frac 85.4%

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

      4. associate-*r* 80.8%

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

      5. sqrt-prod 80.8%

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

      6. sqrt-prod 45.0%

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

      7. add-sqr-sqrt 74.8%

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

      8. sqrt-prod 42.0%

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

      9. add-sqr-sqrt 75.0%

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

      10. associate-*r* 67.3%

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

      11. sqrt-prod 67.3%

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

      12. sqrt-prod 38.6%

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

      13. add-sqr-sqrt 73.5%

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

      14. sqrt-prod 50.1%

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

      15. add-sqr-sqrt 91.5%

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

   7. Applied egg-rr 91.5%

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

   if 1.82e21 < v

   1. Initial program 76.2%

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

      1. associate--l- 76.2%

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

      2. associate-*l* 69.1%

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

      3. sqr-neg 69.1%

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

      4. associate-*l* 76.2%

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

      5. associate-/l* 87.1%

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

      6. fma-define 87.1%

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

   3. Simplified 87.1%

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

      1. add-sqr-sqrt 87.1%

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

      2. *-un-lft-identity 87.1%

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

      3. times-frac 87.0%

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

      4. associate-*r* 78.4%

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

      5. sqrt-prod 78.5%

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

      6. sqrt-prod 48.6%

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

      7. add-sqr-sqrt 62.1%

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

      8. sqrt-prod 41.8%

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

      9. add-sqr-sqrt 67.3%

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

      10. associate-*r* 58.5%

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

      11. sqrt-prod 58.5%

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

      12. sqrt-prod 42.1%

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

      13. add-sqr-sqrt 68.9%

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

      14. sqrt-prod 59.9%

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

      15. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 71.7%

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

      1. /-rgt-identity 71.7%

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

      2. *-commutative 71.7%

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

   9. Applied egg-rr 71.7%

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

   10. Taylor expanded in v around inf 71.7%

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

       1. mul-1-neg 71.7%

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

   12. Simplified 71.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.82 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= v 5.8e+15)
   (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (/ (* r (* r (* w w))) (+ v -1.0)))))
   (+ 3.0 (- (* (* 0.125 (+ 3.0 (* v -2.0))) (* (* r w) (/ (* r w) v))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (v <= 5.8e+15) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (v <= 5.8d+15) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * ((r * (r * (w * w))) / (v + (-1.0d0)))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * ((r * w) * ((r * w) / v))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (v <= 5.8e+15) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * (r * (w * w))) / (v + -1.0))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if v <= 5.8e+15:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * (r * (w * w))) / (v + -1.0))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (v <= 5.8e+15)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(v + -1.0)))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(Float64(r * w) * Float64(Float64(r * w) / v))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (v <= 5.8e+15)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * (r * (w * w))) / (v + -1.0))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[v, 5.8e+15], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 5.8e15

   1. Initial program 87.5%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in v around 0 85.4%

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

   if 5.8e15 < v

   1. Initial program 76.2%

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

      1. associate--l- 76.2%

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

      2. associate-*l* 69.1%

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

      3. sqr-neg 69.1%

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

      4. associate-*l* 76.2%

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

      5. associate-/l* 87.1%

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

      6. fma-define 87.1%

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

   3. Simplified 87.1%

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

      1. add-sqr-sqrt 87.1%

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

      2. *-un-lft-identity 87.1%

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

      3. times-frac 87.0%

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

      4. associate-*r* 78.4%

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

      5. sqrt-prod 78.5%

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

      6. sqrt-prod 48.6%

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

      7. add-sqr-sqrt 62.1%

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

      8. sqrt-prod 41.8%

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

      9. add-sqr-sqrt 67.3%

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

      10. associate-*r* 58.5%

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

      11. sqrt-prod 58.5%

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

      12. sqrt-prod 42.1%

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

      13. add-sqr-sqrt 68.9%

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

      14. sqrt-prod 59.9%

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

      15. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 71.7%

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

      1. /-rgt-identity 71.7%

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

      2. *-commutative 71.7%

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

   9. Applied egg-rr 71.7%

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

   10. Taylor expanded in v around inf 71.7%

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

       1. mul-1-neg 71.7%

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

   12. Simplified 71.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 6.2 \cdot 10^{-121}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 6.2e-121)
   (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5)
   (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (/ (* r (* r (* w w))) (+ v -1.0)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 6.2e-121) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * (r * (w * w))) / (v + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 6.2d-121) then
        tmp = (3.0d0 + ((2.0d0 / r) / r)) - 4.5d0
    else
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * ((r * (r * (w * w))) / (v + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 6.2e-121) {
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * (r * (w * w))) / (v + -1.0))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 6.2e-121:
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5
	else:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * (r * (w * w))) / (v + -1.0))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 6.2e-121)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) - 4.5);
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(v + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 6.2e-121)
		tmp = (3.0 + ((2.0 / r) / r)) - 4.5;
	else
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * ((r * (r * (w * w))) / (v + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 6.2e-121], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 6.1999999999999997e-121

   1. Initial program 82.3%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in r around 0 68.5%

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

      1. associate-/r* 68.5%

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

      2. div-inv 68.5%

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

   7. Applied egg-rr 68.5%

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

      1. associate-*r/ 68.5%

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

      2. *-rgt-identity 68.5%

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

   9. Simplified 68.5%

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

   if 6.1999999999999997e-121 < r

   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 93.6%

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

   5. Taylor expanded in v around 0 74.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{0.375} \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.2 \cdot 10^{-121}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (2.0d0 / (r * r)) + ((-1.5d0) + ((0.375d0 + ((-0.25d0) * v)) / ((v + (-1.0d0)) / ((r * w) * (r * w)))))
end function

Java

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

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (-0.25 * v)) / ((v + -1.0) / ((r * w) * (r * w)))))

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + ((0.375 + (-0.25 * v)) / ((v + -1.0) / ((r * w) * (r * w)))));
end

Wolfram

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

Derivation

1. Initial program 84.9%

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

3. Simplified 89.3%

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

   1. associate-*r/ 90.1%

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

   2. *-commutative 90.1%

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

   3. associate-/l* 90.3%

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

   4. clear-num 90.3%

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

   5. un-div-inv 90.3%

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

   6. fma-undefine 90.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   7. *-commutative 90.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   8. distribute-lft-in 90.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.125 \cdot \left(-2 \cdot v\right) + 0.125 \cdot 3}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   9. metadata-eval 90.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \left(-2 \cdot v\right) + \color{blue}{0.375}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   10. associate-*r* 90.3%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(0.125 \cdot -2\right) \cdot v} + 0.375}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. metadata-eval 90.3%

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

   12. associate-*r* 83.7%

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

   13. pow2 83.7%

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

   14. pow2 83.7%

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

   15. pow-prod-down 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 8: 56.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

3. Simplified 81.3%

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

5. Taylor expanded in r around 0 60.3%

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

6. Final simplification 60.3%

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

Alternative 9: 14.4% 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 84.9%

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

3. Simplified 81.3%

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

5. Taylor expanded in r around 0 60.3%

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

6. Taylor expanded in r around inf 19.7%

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

Reproduce

herbie shell --seed 2024177 
(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))