Rosa's TurbineBenchmark

Percentage Accurate: 84.4% → 99.8%

Time: 10.9s

Alternatives: 8

Speedup: 1.7×

• 52.8% 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: 84.4% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Simplified 86.7%

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

4. Taylor expanded in r around 0 82.2%

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

   1. unpow2 82.2%

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

   2. unpow2 82.2%

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

   3. swap-sqr 99.8%

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

   4. unpow2 99.8%

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

6. Simplified 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -3.1 \cdot 10^{+26}:\\ \;\;\;\;-1.5 + \left(t_0 + t_1 \cdot -0.25\right)\\ \mathbf{elif}\;v \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;-1.5 + \left(t_0 + -0.375 \cdot \frac{r \cdot w}{\frac{\frac{1}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + t_1 \cdot \left(-0.25 + \frac{0.125}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* (* r w) (* r w))))
   (if (<= v -3.1e+26)
     (+ -1.5 (+ t_0 (* t_1 -0.25)))
     (if (<= v 3.8e-5)
       (+ -1.5 (+ t_0 (* -0.375 (/ (* r w) (/ (/ 1.0 r) w)))))
       (+ -1.5 (+ t_0 (* t_1 (+ -0.25 (/ 0.125 v)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (r * w) * (r * w);
	double tmp;
	if (v <= -3.1e+26) {
		tmp = -1.5 + (t_0 + (t_1 * -0.25));
	} else if (v <= 3.8e-5) {
		tmp = -1.5 + (t_0 + (-0.375 * ((r * w) / ((1.0 / r) / w))));
	} else {
		tmp = -1.5 + (t_0 + (t_1 * (-0.25 + (0.125 / v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (r * w) * (r * w)
    if (v <= (-3.1d+26)) then
        tmp = (-1.5d0) + (t_0 + (t_1 * (-0.25d0)))
    else if (v <= 3.8d-5) then
        tmp = (-1.5d0) + (t_0 + ((-0.375d0) * ((r * w) / ((1.0d0 / r) / w))))
    else
        tmp = (-1.5d0) + (t_0 + (t_1 * ((-0.25d0) + (0.125d0 / v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (r * w) * (r * w);
	double tmp;
	if (v <= -3.1e+26) {
		tmp = -1.5 + (t_0 + (t_1 * -0.25));
	} else if (v <= 3.8e-5) {
		tmp = -1.5 + (t_0 + (-0.375 * ((r * w) / ((1.0 / r) / w))));
	} else {
		tmp = -1.5 + (t_0 + (t_1 * (-0.25 + (0.125 / v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (r * w) * (r * w)
	tmp = 0
	if v <= -3.1e+26:
		tmp = -1.5 + (t_0 + (t_1 * -0.25))
	elif v <= 3.8e-5:
		tmp = -1.5 + (t_0 + (-0.375 * ((r * w) / ((1.0 / r) / w))))
	else:
		tmp = -1.5 + (t_0 + (t_1 * (-0.25 + (0.125 / v))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(r * w) * Float64(r * w))
	tmp = 0.0
	if (v <= -3.1e+26)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(t_1 * -0.25)));
	elseif (v <= 3.8e-5)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(-0.375 * Float64(Float64(r * w) / Float64(Float64(1.0 / r) / w)))));
	else
		tmp = Float64(-1.5 + Float64(t_0 + Float64(t_1 * Float64(-0.25 + Float64(0.125 / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (r * w) * (r * w);
	tmp = 0.0;
	if (v <= -3.1e+26)
		tmp = -1.5 + (t_0 + (t_1 * -0.25));
	elseif (v <= 3.8e-5)
		tmp = -1.5 + (t_0 + (-0.375 * ((r * w) / ((1.0 / r) / w))));
	else
		tmp = -1.5 + (t_0 + (t_1 * (-0.25 + (0.125 / v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -3.1e+26], N[(-1.5 + N[(t$95$0 + N[(t$95$1 * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 3.8e-5], N[(-1.5 + N[(t$95$0 + N[(-0.375 * N[(N[(r * w), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$0 + N[(t$95$1 * N[(-0.25 + N[(0.125 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.1e26

   1. Initial program 78.1%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in r around 0 81.8%

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

      1. unpow2 81.8%

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

      2. unpow2 81.8%

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

      3. swap-sqr 99.7%

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

      4. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in v around inf 99.7%

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

   if -3.1e26 < v < 3.8000000000000002e-5

   1. Initial program 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in v around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. swap-sqr 99.2%

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

      5. unpow2 99.2%

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

   6. Simplified 99.2%

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

      1. unpow2 99.2%

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

      2. remove-double-div 99.2%

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

      3. un-div-inv 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. inv-pow 99.2%

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

      2. *-commutative 99.2%

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

      3. unpow-prod-down 99.2%

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

      4. inv-pow 99.2%

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

      5. inv-pow 99.2%

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

   10. Applied egg-rr 99.2%

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

   11. Taylor expanded in w around 0 99.2%

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

       1. associate-/r* 99.2%

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

   13. Simplified 99.2%

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

   if 3.8000000000000002e-5 < v

   1. Initial program 76.4%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in r around 0 80.4%

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

      1. unpow2 80.4%

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

      2. unpow2 80.4%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around inf 98.7%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(0.125 \cdot \frac{1}{v} - 0.25\right)} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -1.5 \]
   10. Step-by-step derivation

       1. sub-neg 98.7%

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

       2. associate-*r/ 98.7%

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

       3. metadata-eval 98.7%

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

       4. metadata-eval 98.7%

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

   11. Simplified 98.7%

       \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\frac{0.125}{v} + -0.25\right)} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -1.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.1 \cdot 10^{+26}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{elif}\;v \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \frac{r \cdot w}{\frac{\frac{1}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(-0.25 + \frac{0.125}{v}\right)\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -1:\\ \;\;\;\;-1.5 + \left(t_0 + t_1 \cdot -0.25\right)\\ \mathbf{elif}\;v \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;-1.5 + \left(t_0 + t_1 \cdot \left(v \cdot -0.125 - 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + t_1 \cdot \left(-0.25 + \frac{0.125}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* (* r w) (* r w))))
   (if (<= v -1.0)
     (+ -1.5 (+ t_0 (* t_1 -0.25)))
     (if (<= v 3.8e-5)
       (+ -1.5 (+ t_0 (* t_1 (- (* v -0.125) 0.375))))
       (+ -1.5 (+ t_0 (* t_1 (+ -0.25 (/ 0.125 v)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (r * w) * (r * w);
	double tmp;
	if (v <= -1.0) {
		tmp = -1.5 + (t_0 + (t_1 * -0.25));
	} else if (v <= 3.8e-5) {
		tmp = -1.5 + (t_0 + (t_1 * ((v * -0.125) - 0.375)));
	} else {
		tmp = -1.5 + (t_0 + (t_1 * (-0.25 + (0.125 / v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (r * w) * (r * w)
    if (v <= (-1.0d0)) then
        tmp = (-1.5d0) + (t_0 + (t_1 * (-0.25d0)))
    else if (v <= 3.8d-5) then
        tmp = (-1.5d0) + (t_0 + (t_1 * ((v * (-0.125d0)) - 0.375d0)))
    else
        tmp = (-1.5d0) + (t_0 + (t_1 * ((-0.25d0) + (0.125d0 / v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (r * w) * (r * w);
	double tmp;
	if (v <= -1.0) {
		tmp = -1.5 + (t_0 + (t_1 * -0.25));
	} else if (v <= 3.8e-5) {
		tmp = -1.5 + (t_0 + (t_1 * ((v * -0.125) - 0.375)));
	} else {
		tmp = -1.5 + (t_0 + (t_1 * (-0.25 + (0.125 / v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (r * w) * (r * w)
	tmp = 0
	if v <= -1.0:
		tmp = -1.5 + (t_0 + (t_1 * -0.25))
	elif v <= 3.8e-5:
		tmp = -1.5 + (t_0 + (t_1 * ((v * -0.125) - 0.375)))
	else:
		tmp = -1.5 + (t_0 + (t_1 * (-0.25 + (0.125 / v))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(r * w) * Float64(r * w))
	tmp = 0.0
	if (v <= -1.0)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(t_1 * -0.25)));
	elseif (v <= 3.8e-5)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(t_1 * Float64(Float64(v * -0.125) - 0.375))));
	else
		tmp = Float64(-1.5 + Float64(t_0 + Float64(t_1 * Float64(-0.25 + Float64(0.125 / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (r * w) * (r * w);
	tmp = 0.0;
	if (v <= -1.0)
		tmp = -1.5 + (t_0 + (t_1 * -0.25));
	elseif (v <= 3.8e-5)
		tmp = -1.5 + (t_0 + (t_1 * ((v * -0.125) - 0.375)));
	else
		tmp = -1.5 + (t_0 + (t_1 * (-0.25 + (0.125 / v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1.0], N[(-1.5 + N[(t$95$0 + N[(t$95$1 * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 3.8e-5], N[(-1.5 + N[(t$95$0 + N[(t$95$1 * N[(N[(v * -0.125), $MachinePrecision] - 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$0 + N[(t$95$1 * N[(-0.25 + N[(0.125 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1

   1. Initial program 79.8%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in r around 0 83.2%

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

      1. unpow2 83.2%

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

      2. unpow2 83.2%

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

      3. swap-sqr 99.7%

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

      4. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in v around inf 99.7%

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

   if -1 < v < 3.8000000000000002e-5

   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 87.8%

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

   4. Taylor expanded in r around 0 82.5%

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

      1. unpow2 82.5%

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

      2. unpow2 82.5%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around 0 99.4%

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

   if 3.8000000000000002e-5 < v

   1. Initial program 76.4%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in r around 0 80.4%

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

      1. unpow2 80.4%

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

      2. unpow2 80.4%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around inf 98.7%

      \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(0.125 \cdot \frac{1}{v} - 0.25\right)} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -1.5 \]
   10. Step-by-step derivation

       1. sub-neg 98.7%

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

       2. associate-*r/ 98.7%

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

       3. metadata-eval 98.7%

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

       4. metadata-eval 98.7%

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

   11. Simplified 98.7%

       \[\leadsto \left(\frac{2}{r \cdot r} + \color{blue}{\left(\frac{0.125}{v} + -0.25\right)} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -1.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{elif}\;v \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(v \cdot -0.125 - 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(-0.25 + \frac{0.125}{v}\right)\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3.2 \cdot 10^{+26} \lor \neg \left(v \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;-1.5 + \left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + -0.375 \cdot \frac{r \cdot w}{\frac{1}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -3.2e+26) (not (<= v 3.8e-5)))
     (+ -1.5 (+ t_0 (* (* (* r w) (* r w)) -0.25)))
     (+ -1.5 (+ t_0 (* -0.375 (/ (* 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 <= -3.2e+26) || !(v <= 3.8e-5)) {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	} else {
		tmp = -1.5 + (t_0 + (-0.375 * ((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 <= (-3.2d+26)) .or. (.not. (v <= 3.8d-5))) then
        tmp = (-1.5d0) + (t_0 + (((r * w) * (r * w)) * (-0.25d0)))
    else
        tmp = (-1.5d0) + (t_0 + ((-0.375d0) * ((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 <= -3.2e+26) || !(v <= 3.8e-5)) {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	} else {
		tmp = -1.5 + (t_0 + (-0.375 * ((r * w) / (1.0 / (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -3.2e+26) or not (v <= 3.8e-5):
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25))
	else:
		tmp = -1.5 + (t_0 + (-0.375 * ((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 <= -3.2e+26) || !(v <= 3.8e-5))
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25)));
	else
		tmp = Float64(-1.5 + Float64(t_0 + Float64(-0.375 * Float64(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 <= -3.2e+26) || ~((v <= 3.8e-5)))
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	else
		tmp = -1.5 + (t_0 + (-0.375 * ((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, -3.2e+26], N[Not[LessEqual[v, 3.8e-5]], $MachinePrecision]], N[(-1.5 + N[(t$95$0 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$0 + N[(-0.375 * N[(N[(r * w), $MachinePrecision] / N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.20000000000000029e26 or 3.8000000000000002e-5 < v

   1. Initial program 77.3%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in r around 0 81.1%

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

      1. unpow2 81.1%

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

      2. unpow2 81.1%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around inf 99.2%

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

   if -3.20000000000000029e26 < v < 3.8000000000000002e-5

   1. Initial program 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in v around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. swap-sqr 99.2%

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

      5. unpow2 99.2%

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

   6. Simplified 99.2%

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

      1. unpow2 99.2%

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

      2. remove-double-div 99.2%

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

      3. un-div-inv 99.2%

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

   8. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.2 \cdot 10^{+26} \lor \neg \left(v \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \frac{r \cdot w}{\frac{1}{r \cdot w}}\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3.1 \cdot 10^{+26} \lor \neg \left(v \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;-1.5 + \left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + -0.375 \cdot \frac{r \cdot w}{\frac{\frac{1}{r}}{w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -3.1e+26) (not (<= v 5e-9)))
     (+ -1.5 (+ t_0 (* (* (* r w) (* r w)) -0.25)))
     (+ -1.5 (+ t_0 (* -0.375 (/ (* 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 <= -3.1e+26) || !(v <= 5e-9)) {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	} else {
		tmp = -1.5 + (t_0 + (-0.375 * ((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 <= (-3.1d+26)) .or. (.not. (v <= 5d-9))) then
        tmp = (-1.5d0) + (t_0 + (((r * w) * (r * w)) * (-0.25d0)))
    else
        tmp = (-1.5d0) + (t_0 + ((-0.375d0) * ((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 <= -3.1e+26) || !(v <= 5e-9)) {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	} else {
		tmp = -1.5 + (t_0 + (-0.375 * ((r * w) / ((1.0 / r) / w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -3.1e+26) or not (v <= 5e-9):
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25))
	else:
		tmp = -1.5 + (t_0 + (-0.375 * ((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 <= -3.1e+26) || !(v <= 5e-9))
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25)));
	else
		tmp = Float64(-1.5 + Float64(t_0 + Float64(-0.375 * Float64(Float64(r * w) / Float64(Float64(1.0 / 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 <= -3.1e+26) || ~((v <= 5e-9)))
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	else
		tmp = -1.5 + (t_0 + (-0.375 * ((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, -3.1e+26], N[Not[LessEqual[v, 5e-9]], $MachinePrecision]], N[(-1.5 + N[(t$95$0 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$0 + N[(-0.375 * N[(N[(r * w), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.1e26 or 5.0000000000000001e-9 < v

   1. Initial program 77.3%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in r around 0 81.1%

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

      1. unpow2 81.1%

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

      2. unpow2 81.1%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around inf 99.2%

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

   if -3.1e26 < v < 5.0000000000000001e-9

   1. Initial program 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in v around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. swap-sqr 99.2%

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

      5. unpow2 99.2%

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

   6. Simplified 99.2%

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

      1. unpow2 99.2%

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

      2. remove-double-div 99.2%

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

      3. un-div-inv 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. inv-pow 99.2%

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

      2. *-commutative 99.2%

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

      3. unpow-prod-down 99.2%

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

      4. inv-pow 99.2%

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

      5. inv-pow 99.2%

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

   10. Applied egg-rr 99.2%

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

   11. Taylor expanded in w around 0 99.2%

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

       1. associate-/r* 99.2%

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

   13. Simplified 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.1 \cdot 10^{+26} \lor \neg \left(v \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \frac{r \cdot w}{\frac{\frac{1}{r}}{w}}\right)\\ \end{array} \]

Alternative 6: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1 \cdot 10^{+36} \lor \neg \left(v \leq 3.2 \cdot 10^{-5}\right):\\ \;\;\;\;-1.5 + \left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + -0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1e+36) (not (<= v 3.2e-5)))
     (+ -1.5 (+ t_0 (* (* (* r w) (* r w)) -0.25)))
     (+ -1.5 (+ t_0 (* -0.375 (* r (* w (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1e+36) || !(v <= 3.2e-5)) {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	} else {
		tmp = -1.5 + (t_0 + (-0.375 * (r * (w * (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 <= (-1d+36)) .or. (.not. (v <= 3.2d-5))) then
        tmp = (-1.5d0) + (t_0 + (((r * w) * (r * w)) * (-0.25d0)))
    else
        tmp = (-1.5d0) + (t_0 + ((-0.375d0) * (r * (w * (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 <= -1e+36) || !(v <= 3.2e-5)) {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	} else {
		tmp = -1.5 + (t_0 + (-0.375 * (r * (w * (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1e+36) or not (v <= 3.2e-5):
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25))
	else:
		tmp = -1.5 + (t_0 + (-0.375 * (r * (w * (r * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1e+36) || !(v <= 3.2e-5))
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25)));
	else
		tmp = Float64(-1.5 + Float64(t_0 + Float64(-0.375 * Float64(r * Float64(w * 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 <= -1e+36) || ~((v <= 3.2e-5)))
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	else
		tmp = -1.5 + (t_0 + (-0.375 * (r * (w * (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, -1e+36], N[Not[LessEqual[v, 3.2e-5]], $MachinePrecision]], N[(-1.5 + N[(t$95$0 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$0 + N[(-0.375 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.00000000000000004e36 or 3.19999999999999986e-5 < v

   1. Initial program 77.3%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in r around 0 81.1%

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

      1. unpow2 81.1%

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

      2. unpow2 81.1%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around inf 99.2%

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

   if -1.00000000000000004e36 < v < 3.19999999999999986e-5

   1. Initial program 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in v around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. swap-sqr 99.2%

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

      5. unpow2 99.2%

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

   6. Simplified 99.2%

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

      1. unpow2 99.2%

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

      2. remove-double-div 99.2%

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

      3. un-div-inv 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. associate-/r/ 99.2%

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

      2. /-rgt-identity 99.2%

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

      3. associate-*l* 96.1%

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

   10. Applied egg-rr 96.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+36} \lor \neg \left(v \leq 3.2 \cdot 10^{-5}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2 \cdot 10^{+31} \lor \neg \left(v \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;-1.5 + \left(t_1 + t_0 \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_1 + -0.375 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -2e+31) (not (<= v 3.8e-5)))
     (+ -1.5 (+ t_1 (* t_0 -0.25)))
     (+ -1.5 (+ t_1 (* -0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2e+31) || !(v <= 3.8e-5)) {
		tmp = -1.5 + (t_1 + (t_0 * -0.25));
	} else {
		tmp = -1.5 + (t_1 + (-0.375 * 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 = (r * w) * (r * w)
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-2d+31)) .or. (.not. (v <= 3.8d-5))) then
        tmp = (-1.5d0) + (t_1 + (t_0 * (-0.25d0)))
    else
        tmp = (-1.5d0) + (t_1 + ((-0.375d0) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2e+31) || !(v <= 3.8e-5)) {
		tmp = -1.5 + (t_1 + (t_0 * -0.25));
	} else {
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2e+31) or not (v <= 3.8e-5):
		tmp = -1.5 + (t_1 + (t_0 * -0.25))
	else:
		tmp = -1.5 + (t_1 + (-0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2e+31) || !(v <= 3.8e-5))
		tmp = Float64(-1.5 + Float64(t_1 + Float64(t_0 * -0.25)));
	else
		tmp = Float64(-1.5 + Float64(t_1 + Float64(-0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2e+31) || ~((v <= 3.8e-5)))
		tmp = -1.5 + (t_1 + (t_0 * -0.25));
	else
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -2e+31], N[Not[LessEqual[v, 3.8e-5]], $MachinePrecision]], N[(-1.5 + N[(t$95$1 + N[(t$95$0 * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$1 + N[(-0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.9999999999999999e31 or 3.8000000000000002e-5 < v

   1. Initial program 77.3%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in r around 0 81.1%

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

      1. unpow2 81.1%

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

      2. unpow2 81.1%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around inf 99.2%

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

   if -1.9999999999999999e31 < v < 3.8000000000000002e-5

   1. Initial program 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in v around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. swap-sqr 99.2%

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

      5. unpow2 99.2%

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

   6. Simplified 99.2%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+31} \lor \neg \left(v \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 8: 93.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return -1.5 + ((2.0 / (r * r)) + (((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 = (-1.5d0) + ((2.0d0 / (r * r)) + (((r * w) * (r * w)) * (-0.25d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Simplified 86.7%

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

4. Taylor expanded in r around 0 82.2%

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

   1. unpow2 82.2%

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

   2. unpow2 82.2%

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

   3. swap-sqr 99.8%

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

   4. unpow2 99.8%

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

6. Simplified 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in v around inf 94.8%

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

10. Final simplification 94.8%

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

Reproduce

herbie shell --seed 2023306 
(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))