Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 99.8%

Time: 14.3s

Alternatives: 5

Speedup: 1.7×

• 53.5% 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

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

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(Float64(2.0 / 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[(N[(2.0 / r), $MachinePrecision] / r), $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 87.8%

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

3. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. frac-2neg 97.6%

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

   2. *-commutative 97.6%

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

   3. associate-*r* 89.0%

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

   4. div-inv 88.9%

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

   5. associate-*r* 97.6%

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

   6. *-commutative 97.6%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   4. fma-udef 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   5. *-commutative 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   6. +-commutative 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   7. associate--r+ 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{0 - \left(1 - v\right)}}{{\left(w \cdot r\right)}^{2}}}\right) \]

   12. associate--r- 99.8%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{\left(0 - 1\right) + v}}{{\left(w \cdot r\right)}^{2}}}\right) \]

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. unpow2 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\\ t_1 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -27000000:\\ \;\;\;\;t\_0 + \frac{-2}{r \cdot \left(-r\right)}\\ \mathbf{elif}\;v \leq 8.6 \cdot 10^{-91}:\\ \;\;\;\;t\_1 + \left(-1.5 + \frac{v \cdot -0.25 - -0.375}{\frac{\frac{-1}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- -1.5 (* (* (* r w) (* r w)) 0.25))) (t_1 (/ (/ 2.0 r) r)))
   (if (<= v -27000000.0)
     (+ t_0 (/ -2.0 (* r (- r))))
     (if (<= v 8.6e-91)
       (+ t_1 (+ -1.5 (/ (- (* v -0.25) -0.375) (/ (/ -1.0 (* r w)) (* r w)))))
       (+ t_1 t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = -1.5 - (((r * w) * (r * w)) * 0.25);
	double t_1 = (2.0 / r) / r;
	double tmp;
	if (v <= -27000000.0) {
		tmp = t_0 + (-2.0 / (r * -r));
	} else if (v <= 8.6e-91) {
		tmp = t_1 + (-1.5 + (((v * -0.25) - -0.375) / ((-1.0 / (r * w)) / (r * w))));
	} else {
		tmp = t_1 + t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.5d0) - (((r * w) * (r * w)) * 0.25d0)
    t_1 = (2.0d0 / r) / r
    if (v <= (-27000000.0d0)) then
        tmp = t_0 + ((-2.0d0) / (r * -r))
    else if (v <= 8.6d-91) then
        tmp = t_1 + ((-1.5d0) + (((v * (-0.25d0)) - (-0.375d0)) / (((-1.0d0) / (r * w)) / (r * w))))
    else
        tmp = t_1 + t_0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = -1.5 - (((r * w) * (r * w)) * 0.25);
	double t_1 = (2.0 / r) / r;
	double tmp;
	if (v <= -27000000.0) {
		tmp = t_0 + (-2.0 / (r * -r));
	} else if (v <= 8.6e-91) {
		tmp = t_1 + (-1.5 + (((v * -0.25) - -0.375) / ((-1.0 / (r * w)) / (r * w))));
	} else {
		tmp = t_1 + t_0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = -1.5 - (((r * w) * (r * w)) * 0.25)
	t_1 = (2.0 / r) / r
	tmp = 0
	if v <= -27000000.0:
		tmp = t_0 + (-2.0 / (r * -r))
	elif v <= 8.6e-91:
		tmp = t_1 + (-1.5 + (((v * -0.25) - -0.375) / ((-1.0 / (r * w)) / (r * w))))
	else:
		tmp = t_1 + t_0
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25))
	t_1 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if (v <= -27000000.0)
		tmp = Float64(t_0 + Float64(-2.0 / Float64(r * Float64(-r))));
	elseif (v <= 8.6e-91)
		tmp = Float64(t_1 + Float64(-1.5 + Float64(Float64(Float64(v * -0.25) - -0.375) / Float64(Float64(-1.0 / Float64(r * w)) / Float64(r * w)))));
	else
		tmp = Float64(t_1 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = -1.5 - (((r * w) * (r * w)) * 0.25);
	t_1 = (2.0 / r) / r;
	tmp = 0.0;
	if (v <= -27000000.0)
		tmp = t_0 + (-2.0 / (r * -r));
	elseif (v <= 8.6e-91)
		tmp = t_1 + (-1.5 + (((v * -0.25) - -0.375) / ((-1.0 / (r * w)) / (r * w))));
	else
		tmp = t_1 + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[LessEqual[v, -27000000.0], N[(t$95$0 + N[(-2.0 / N[(r * (-r)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 8.6e-91], N[(t$95$1 + N[(-1.5 + N[(N[(N[(v * -0.25), $MachinePrecision] - -0.375), $MachinePrecision] / N[(N[(-1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.7e7

   1. Initial program 91.7%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 86.2%

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

      1. *-commutative 86.2%

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

      2. *-commutative 86.2%

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

      3. unpow2 86.2%

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

      4. unpow2 86.2%

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

      5. swap-sqr 99.6%

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

      6. unpow2 99.6%

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

      7. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.6%

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

       1. div-inv 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. frac-2neg 99.6%

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

       2. metadata-eval 99.6%

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

       3. frac-times 99.6%

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

       4. metadata-eval 99.6%

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

   13. Applied egg-rr 99.6%

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

   if -2.7e7 < v < 8.6e-91

   1. Initial program 86.5%

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

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*r* 86.5%

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

      4. div-inv 86.5%

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

      5. associate-*r* 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*r* 99.8%

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

      8. pow2 99.8%

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

      9. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. neg-sub0 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      4. fma-udef 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      5. *-commutative 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      7. associate--r+ 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-neg-frac 99.8%

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

      11. neg-sub0 99.8%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{0 - \left(1 - v\right)}}{{\left(w \cdot r\right)}^{2}}}\right) \]

      12. associate--r- 99.8%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{\left(0 - 1\right) + v}}{{\left(w \cdot r\right)}^{2}}}\right) \]

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

       3. +-commutative 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in v around 0 99.0%

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

   if 8.6e-91 < v

   1. Initial program 86.9%

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

   3. Simplified 95.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 81.4%

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

      1. *-commutative 81.4%

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

      2. *-commutative 81.4%

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

      3. unpow2 81.4%

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

      4. unpow2 81.4%

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

      5. swap-sqr 98.4%

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

      6. unpow2 98.4%

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

      7. *-commutative 98.4%

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

   7. Simplified 98.4%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 98.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -27000000:\\ \;\;\;\;\left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right) + \frac{-2}{r \cdot \left(-r\right)}\\ \mathbf{elif}\;v \leq 8.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \frac{v \cdot -0.25 - -0.375}{\frac{\frac{-1}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 4.3 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{v \cdot -0.25 - -0.375}{\frac{v + -1}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= w 4.3e+176)
   (+
    (/ (/ 2.0 r) r)
    (+ -1.5 (* (* w (* r w)) (/ (- (* v -0.25) -0.375) (/ (+ v -1.0) r)))))
   (+ (* (/ 2.0 r) (/ 1.0 r)) (- -1.5 (* (* (* r w) (* r w)) 0.25)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (w <= 4.3e+176) {
		tmp = ((2.0 / r) / r) + (-1.5 + ((w * (r * w)) * (((v * -0.25) - -0.375) / ((v + -1.0) / r))));
	} else {
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	}
	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 (w <= 4.3d+176) then
        tmp = ((2.0d0 / r) / r) + ((-1.5d0) + ((w * (r * w)) * (((v * (-0.25d0)) - (-0.375d0)) / ((v + (-1.0d0)) / r))))
    else
        tmp = ((2.0d0 / r) * (1.0d0 / r)) + ((-1.5d0) - (((r * w) * (r * w)) * 0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (w <= 4.3e+176) {
		tmp = ((2.0 / r) / r) + (-1.5 + ((w * (r * w)) * (((v * -0.25) - -0.375) / ((v + -1.0) / r))));
	} else {
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if w <= 4.3e+176:
		tmp = ((2.0 / r) / r) + (-1.5 + ((w * (r * w)) * (((v * -0.25) - -0.375) / ((v + -1.0) / r))))
	else:
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (((r * w) * (r * w)) * 0.25))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (w <= 4.3e+176)
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 + Float64(Float64(w * Float64(r * w)) * Float64(Float64(Float64(v * -0.25) - -0.375) / Float64(Float64(v + -1.0) / r)))));
	else
		tmp = Float64(Float64(Float64(2.0 / r) * Float64(1.0 / r)) + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (w <= 4.3e+176)
		tmp = ((2.0 / r) / r) + (-1.5 + ((w * (r * w)) * (((v * -0.25) - -0.375) / ((v + -1.0) / r))));
	else
		tmp = ((2.0 / r) * (1.0 / r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[w, 4.3e+176], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 + N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(v * -0.25), $MachinePrecision] - -0.375), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 4.30000000000000026e176

   1. Initial program 90.5%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*r* 91.8%

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

      4. div-inv 91.7%

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

      5. associate-*r* 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-*r* 99.8%

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

      8. pow2 99.8%

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

      9. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. neg-sub0 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      4. fma-udef 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      5. *-commutative 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      7. associate--r+ 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}\right) \]

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-neg-frac 99.8%

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

      11. neg-sub0 99.8%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{0 - \left(1 - v\right)}}{{\left(w \cdot r\right)}^{2}}}\right) \]

      12. associate--r- 99.8%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{-0.375 - v \cdot -0.25}{\frac{\color{blue}{\left(0 - 1\right) + v}}{{\left(w \cdot r\right)}^{2}}}\right) \]

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

       3. +-commutative 99.8%

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

   12. Simplified 99.8%

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

       1. associate-/r* 99.8%

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

       2. *-rgt-identity 99.8%

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

       3. associate-/r* 99.5%

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

       4. associate-/r/ 99.4%

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

       5. *-rgt-identity 99.4%

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

       6. +-commutative 99.4%

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

   14. Applied egg-rr 99.4%

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

   if 4.30000000000000026e176 < w

   1. Initial program 69.3%

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

   3. Simplified 84.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 69.3%

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

      1. *-commutative 69.3%

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

      2. *-commutative 69.3%

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

      3. unpow2 69.3%

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

      4. unpow2 69.3%

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

      5. swap-sqr 96.4%

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

      6. unpow2 96.4%

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

      7. *-commutative 96.4%

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

   7. Simplified 96.4%

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

      1. unpow2 100.0%

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

   9. Applied egg-rr 96.4%

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

       1. div-inv 96.4%

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

   11. Applied egg-rr 96.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 4.3 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{v \cdot -0.25 - -0.375}{\frac{v + -1}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r} \cdot \frac{1}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

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) - (((r * w) * (r * w)) * 0.25d0)) + ((-2.0d0) / (r * -r))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 80.4%

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

   1. *-commutative 80.4%

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

   2. *-commutative 80.4%

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

   3. unpow2 80.4%

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

   4. unpow2 80.4%

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

   5. swap-sqr 94.5%

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

   6. unpow2 94.5%

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

   7. *-commutative 94.5%

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

7. Simplified 94.5%

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

   1. unpow2 99.8%

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

9. Applied egg-rr 94.5%

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

    1. div-inv 94.4%

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

11. Applied egg-rr 94.4%

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

    1. frac-2neg 94.4%

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

    2. metadata-eval 94.4%

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

    3. frac-times 94.5%

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

    4. metadata-eval 94.5%

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

13. Applied egg-rr 94.5%

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

14. Final simplification 94.5%

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

Alternative 5: 93.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 80.4%

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

   1. *-commutative 80.4%

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

   2. *-commutative 80.4%

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

   3. unpow2 80.4%

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

   4. unpow2 80.4%

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

   5. swap-sqr 94.5%

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

   6. unpow2 94.5%

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

   7. *-commutative 94.5%

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

7. Simplified 94.5%

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

   1. unpow2 99.8%

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

9. Applied egg-rr 94.5%

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

10. Final simplification 94.5%

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

Reproduce

herbie shell --seed 2024027 
(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))