Rosa's TurbineBenchmark

Percentage Accurate: 84.3% → 99.8%

Time: 15.4s

Alternatives: 4

Speedup: 1.5×

• 54.3% 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: 84.3% 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

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

FPCore

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

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / ((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) + (((v * (-0.25d0)) + 0.375d0) / ((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 + (((v * -0.25) + 0.375) / ((v + -1.0) / ((r * w) * (r * w)))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / ((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(Float64(v * -0.25) + 0.375) / 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 + (((v * -0.25) + 0.375) / ((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[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

3. Simplified 86.2%

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

   1. fma-undefine 86.2%

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

   2. *-commutative 86.2%

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

   3. +-commutative 86.2%

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

   4. associate-*r/ 86.6%

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

   5. *-commutative 86.6%

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

   6. associate-/l* 87.4%

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

   7. clear-num 87.4%

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

   8. un-div-inv 87.4%

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

   9. +-commutative 87.4%

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

   10. distribute-rgt-in 87.4%

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

   11. *-commutative 87.4%

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

   12. associate-*l* 87.8%

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

   13. metadata-eval 87.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   14. metadata-eval 87.8%

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

   15. associate-*r* 82.2%

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

   16. pow2 82.2%

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

   17. pow2 82.2%

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

   18. pow-prod-down 99.9%

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

   19. *-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

   1. unpow2 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -170000000 \lor \neg \left(v \leq 0.19\right):\\ \;\;\;\;\left(t\_0 + 3\right) + \left(\left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r \cdot \frac{r \cdot w}{v}\right)\right) - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{1}{r \cdot w} \cdot \frac{-1}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -170000000.0) (not (<= v 0.19)))
     (+ (+ t_0 3.0) (- (* (* v -0.25) (* w (* r (/ (* r w) v)))) 4.5))
     (+
      t_0
      (+
       -1.5
       (/ (+ (* v -0.25) 0.375) (* (/ 1.0 (* r w)) (/ -1.0 (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -170000000.0) || !(v <= 0.19)) {
		tmp = (t_0 + 3.0) + (((v * -0.25) * (w * (r * ((r * w) / v)))) - 4.5);
	} else {
		tmp = t_0 + (-1.5 + (((v * -0.25) + 0.375) / ((1.0 / (r * w)) * (-1.0 / (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 = 2.0d0 / (r * r)
    if ((v <= (-170000000.0d0)) .or. (.not. (v <= 0.19d0))) then
        tmp = (t_0 + 3.0d0) + (((v * (-0.25d0)) * (w * (r * ((r * w) / v)))) - 4.5d0)
    else
        tmp = t_0 + ((-1.5d0) + (((v * (-0.25d0)) + 0.375d0) / ((1.0d0 / (r * w)) * ((-1.0d0) / (r * w)))))
    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 <= -170000000.0) || !(v <= 0.19)) {
		tmp = (t_0 + 3.0) + (((v * -0.25) * (w * (r * ((r * w) / v)))) - 4.5);
	} else {
		tmp = t_0 + (-1.5 + (((v * -0.25) + 0.375) / ((1.0 / (r * w)) * (-1.0 / (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -170000000.0) or not (v <= 0.19):
		tmp = (t_0 + 3.0) + (((v * -0.25) * (w * (r * ((r * w) / v)))) - 4.5)
	else:
		tmp = t_0 + (-1.5 + (((v * -0.25) + 0.375) / ((1.0 / (r * w)) * (-1.0 / (r * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -170000000.0) || !(v <= 0.19))
		tmp = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(v * -0.25) * Float64(w * Float64(r * Float64(Float64(r * w) / v)))) - 4.5));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(Float64(1.0 / Float64(r * w)) * Float64(-1.0 / Float64(r * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -170000000.0) || ~((v <= 0.19)))
		tmp = (t_0 + 3.0) + (((v * -0.25) * (w * (r * ((r * w) / v)))) - 4.5);
	else
		tmp = t_0 + (-1.5 + (((v * -0.25) + 0.375) / ((1.0 / (r * w)) * (-1.0 / (r * w)))));
	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, -170000000.0], N[Not[LessEqual[v, 0.19]], $MachinePrecision]], N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(v * -0.25), $MachinePrecision] * N[(w * N[(r * N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.7e8 or 0.19 < v

   1. Initial program 82.0%

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

      1. associate--l- 82.0%

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

      2. associate-*l* 77.5%

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

      3. sqr-neg 77.5%

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

      4. associate-*l* 82.0%

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

      5. associate-/l* 87.6%

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

      6. fma-define 87.6%

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

   3. Simplified 87.6%

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

      1. associate-/l* 86.0%

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

      2. *-commutative 86.0%

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

      3. associate-*r/ 85.1%

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

      4. *-commutative 85.1%

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

      5. associate-*l* 93.4%

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

      6. associate-*l* 98.1%

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

      7. associate-*r/ 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in v around inf 99.8%

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

      1. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in v around inf 99.8%

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

       1. associate-*r/ 99.8%

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

       2. associate-*r* 99.8%

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

       3. neg-mul-1 99.8%

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

       4. *-commutative 99.8%

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

   12. Simplified 99.8%

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

   if -1.7e8 < v < 0.19

   1. Initial program 87.2%

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

   3. Simplified 87.2%

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

      1. fma-undefine 87.2%

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

      2. *-commutative 87.2%

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

      3. +-commutative 87.2%

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

      4. associate-*r/ 87.2%

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

      5. *-commutative 87.2%

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

      6. associate-/l* 87.2%

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

      7. clear-num 87.2%

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

      8. un-div-inv 87.2%

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

      9. +-commutative 87.2%

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

      10. distribute-rgt-in 87.2%

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

      11. *-commutative 87.2%

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

      12. associate-*l* 87.2%

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

      13. metadata-eval 87.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      14. metadata-eval 87.2%

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

      15. associate-*r* 82.8%

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

      16. pow2 82.8%

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

      17. pow2 82.8%

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

      18. pow-prod-down 99.9%

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

      19. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. unpow2 99.9%

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

      3. times-frac 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around 0 99.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1}{w \cdot r} \cdot \color{blue}{\frac{1}{r \cdot w}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -170000000 \lor \neg \left(v \leq 0.19\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) + \left(\left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r \cdot \frac{r \cdot w}{v}\right)\right) - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{1}{r \cdot w} \cdot \frac{-1}{r \cdot w}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -7000000 \lor \neg \left(v \leq 1.5\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) + \left(\left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r \cdot \frac{r \cdot w}{v}\right)\right) - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) + \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -7000000.0) (not (<= v 1.5)))
   (+
    (+ (/ 2.0 (* r r)) 3.0)
    (- (* (* v -0.25) (* w (* r (/ (* r w) v)))) 4.5))
   (+
    (+ 3.0 (/ (/ 2.0 r) r))
    (- (* 0.375 (* (* r w) (/ (* r w) (+ v -1.0)))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -7000000.0) || !(v <= 1.5)) {
		tmp = ((2.0 / (r * r)) + 3.0) + (((v * -0.25) * (w * (r * ((r * w) / v)))) - 4.5);
	} else {
		tmp = (3.0 + ((2.0 / r) / r)) + ((0.375 * ((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 ((v <= (-7000000.0d0)) .or. (.not. (v <= 1.5d0))) then
        tmp = ((2.0d0 / (r * r)) + 3.0d0) + (((v * (-0.25d0)) * (w * (r * ((r * w) / v)))) - 4.5d0)
    else
        tmp = (3.0d0 + ((2.0d0 / r) / r)) + ((0.375d0 * ((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 ((v <= -7000000.0) || !(v <= 1.5)) {
		tmp = ((2.0 / (r * r)) + 3.0) + (((v * -0.25) * (w * (r * ((r * w) / v)))) - 4.5);
	} else {
		tmp = (3.0 + ((2.0 / r) / r)) + ((0.375 * ((r * w) * ((r * w) / (v + -1.0)))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -7000000.0) or not (v <= 1.5):
		tmp = ((2.0 / (r * r)) + 3.0) + (((v * -0.25) * (w * (r * ((r * w) / v)))) - 4.5)
	else:
		tmp = (3.0 + ((2.0 / r) / r)) + ((0.375 * ((r * w) * ((r * w) / (v + -1.0)))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -7000000.0) || !(v <= 1.5))
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) + Float64(Float64(Float64(v * -0.25) * Float64(w * Float64(r * Float64(Float64(r * w) / v)))) - 4.5));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) + Float64(Float64(0.375 * 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 ((v <= -7000000.0) || ~((v <= 1.5)))
		tmp = ((2.0 / (r * r)) + 3.0) + (((v * -0.25) * (w * (r * ((r * w) / v)))) - 4.5);
	else
		tmp = (3.0 + ((2.0 / r) / r)) + ((0.375 * ((r * w) * ((r * w) / (v + -1.0)))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -7000000.0], N[Not[LessEqual[v, 1.5]], $MachinePrecision]], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] + N[(N[(N[(v * -0.25), $MachinePrecision] * N[(w * N[(r * N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] + N[(N[(0.375 * 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 v < -7e6 or 1.5 < v

   1. Initial program 81.9%

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

      1. associate--l- 81.9%

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

      2. associate-*l* 77.3%

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

      3. sqr-neg 77.3%

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

      4. associate-*l* 81.9%

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

      5. associate-/l* 87.5%

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

      6. fma-define 87.5%

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

   3. Simplified 87.5%

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

      1. associate-/l* 85.9%

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

      2. *-commutative 85.9%

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

      3. associate-*r/ 85.0%

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

      4. *-commutative 85.0%

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

      5. associate-*l* 93.3%

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

      6. associate-*l* 98.1%

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

      7. associate-*r/ 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in v around inf 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in v around inf 99.9%

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

       1. associate-*r/ 99.9%

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

       2. associate-*r* 99.9%

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

       3. neg-mul-1 99.9%

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

       4. *-commutative 99.9%

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

   12. Simplified 99.9%

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

   if -7e6 < v < 1.5

   1. Initial program 87.3%

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

      1. associate--l- 87.3%

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

      2. associate-*l* 82.9%

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

      3. sqr-neg 82.9%

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

      4. associate-*l* 87.3%

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

      5. associate-/l* 87.3%

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

      6. fma-define 87.3%

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

   3. Simplified 87.3%

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

      1. associate-/r* 87.2%

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

      2. div-inv 87.2%

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

   6. Applied egg-rr 87.2%

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

   7. Taylor expanded in v around 0 86.8%

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

      1. un-div-inv 86.8%

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

   9. Applied egg-rr 86.8%

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

       1. associate-/l* 86.8%

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

       2. *-commutative 86.8%

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

       3. associate-*r/ 86.8%

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

       4. associate-*l* 94.5%

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

       5. associate-*r* 99.4%

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

       6. *-commutative 99.4%

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

       7. associate-*r/ 99.4%

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

   11. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -7000000 \lor \neg \left(v \leq 1.5\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) + \left(\left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r \cdot \frac{r \cdot w}{v}\right)\right) - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) + \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. associate--l- 84.8%

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

   2. associate-*l* 80.3%

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

   3. sqr-neg 80.3%

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

   4. associate-*l* 84.8%

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

   5. associate-/l* 87.4%

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

   6. fma-define 87.4%

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

3. Simplified 87.4%

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

   1. associate-/l* 86.6%

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

   2. *-commutative 86.6%

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

   3. associate-*r/ 86.2%

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

   4. *-commutative 86.2%

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

   5. associate-*l* 94.2%

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

   6. associate-*l* 98.0%

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

   7. associate-*r/ 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in v around 0 82.8%

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

8. Taylor expanded in v around 0 92.2%

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

9. Final simplification 92.2%

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

Reproduce

herbie shell --seed 2024089 
(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))