Rosa's TurbineBenchmark

Percentage Accurate: 85.3% → 99.8%

Time: 10.3s

Alternatives: 6

Speedup: 1.7×

• 53.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

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 86.1%

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

3. Simplified 89.4%

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

4. Taylor expanded in r around 0 84.8%

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

   1. unpow2 84.8%

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

   2. unpow2 84.8%

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

   3. swap-sqr 99.8%

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

   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: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{w}}{r}\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -500:\\ \;\;\;\;-1.5 + \left(t_1 + -0.25 \cdot \frac{w}{\frac{t_0}{r}}\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;\left(t_1 + -1.5\right) - \frac{r \cdot \left(w \cdot 0.375\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_1 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 w) r)) (t_1 (/ 2.0 (* r r))))
   (if (<= v -500.0)
     (+ -1.5 (+ t_1 (* -0.25 (/ w (/ t_0 r)))))
     (if (<= v 2.5e-99)
       (- (+ t_1 -1.5) (/ (* r (* w 0.375)) t_0))
       (+ -1.5 (+ t_1 (* (* (* r w) (* r w)) -0.25)))))))

C

double code(double v, double w, double r) {
	double t_0 = (1.0 / w) / r;
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -500.0) {
		tmp = -1.5 + (t_1 + (-0.25 * (w / (t_0 / r))));
	} else if (v <= 2.5e-99) {
		tmp = (t_1 + -1.5) - ((r * (w * 0.375)) / t_0);
	} else {
		tmp = -1.5 + (t_1 + (((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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / w) / r
    t_1 = 2.0d0 / (r * r)
    if (v <= (-500.0d0)) then
        tmp = (-1.5d0) + (t_1 + ((-0.25d0) * (w / (t_0 / r))))
    else if (v <= 2.5d-99) then
        tmp = (t_1 + (-1.5d0)) - ((r * (w * 0.375d0)) / t_0)
    else
        tmp = (-1.5d0) + (t_1 + (((r * w) * (r * w)) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (1.0 / w) / r;
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -500.0) {
		tmp = -1.5 + (t_1 + (-0.25 * (w / (t_0 / r))));
	} else if (v <= 2.5e-99) {
		tmp = (t_1 + -1.5) - ((r * (w * 0.375)) / t_0);
	} else {
		tmp = -1.5 + (t_1 + (((r * w) * (r * w)) * -0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (1.0 / w) / r
	t_1 = 2.0 / (r * r)
	tmp = 0
	if v <= -500.0:
		tmp = -1.5 + (t_1 + (-0.25 * (w / (t_0 / r))))
	elif v <= 2.5e-99:
		tmp = (t_1 + -1.5) - ((r * (w * 0.375)) / t_0)
	else:
		tmp = -1.5 + (t_1 + (((r * w) * (r * w)) * -0.25))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(1.0 / w) / r)
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -500.0)
		tmp = Float64(-1.5 + Float64(t_1 + Float64(-0.25 * Float64(w / Float64(t_0 / r)))));
	elseif (v <= 2.5e-99)
		tmp = Float64(Float64(t_1 + -1.5) - Float64(Float64(r * Float64(w * 0.375)) / t_0));
	else
		tmp = Float64(-1.5 + Float64(t_1 + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (1.0 / w) / r;
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -500.0)
		tmp = -1.5 + (t_1 + (-0.25 * (w / (t_0 / r))));
	elseif (v <= 2.5e-99)
		tmp = (t_1 + -1.5) - ((r * (w * 0.375)) / t_0);
	else
		tmp = -1.5 + (t_1 + (((r * w) * (r * w)) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(1.0 / w), $MachinePrecision] / r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -500.0], N[(-1.5 + N[(t$95$1 + N[(-0.25 * N[(w / N[(t$95$0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2.5e-99], N[(N[(t$95$1 + -1.5), $MachinePrecision] - N[(N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$1 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -500

   1. Initial program 80.6%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in r around 0 86.2%

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

      1. unpow2 86.2%

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

      2. unpow2 86.2%

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

      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 97.9%

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

       1. associate-*r* 98.0%

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

       2. /-rgt-identity 98.0%

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

       3. associate-/r/ 98.0%

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

       4. associate-/l* 97.9%

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

       5. *-commutative 97.9%

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

       6. associate-/l* 98.0%

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

   11. Applied egg-rr 98.0%

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

   if -500 < v < 2.49999999999999985e-99

   1. Initial program 92.4%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in v around 0 86.6%

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

      1. *-commutative 86.6%

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

      2. unpow2 86.6%

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

      3. unpow2 86.6%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

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

      1. associate-*r* 99.0%

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

      2. /-rgt-identity 99.0%

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

      3. associate-/r/ 99.0%

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

      4. clear-num 99.0%

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

      5. clear-num 99.0%

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

      6. associate-/l* 99.8%

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

      7. associate-*l/ 99.8%

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

      8. associate-*l* 99.9%

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

   10. Applied egg-rr 99.9%

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

   if 2.49999999999999985e-99 < v

   1. Initial program 81.9%

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

   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 81.5%

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

      1. unpow2 81.5%

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

      2. unpow2 81.5%

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

      3. swap-sqr 99.9%

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

      4. unpow2 99.9%

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

   6. Simplified 99.9%

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

      1. unpow2 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in v around inf 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -500:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.25 \cdot \frac{w}{\frac{\frac{\frac{1}{w}}{r}}{r}}\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(w \cdot 0.375\right)}{\frac{\frac{1}{w}}{r}}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -480:\\ \;\;\;\;-1.5 + \left(t_0 + t_1 \cdot \left(\frac{0.125}{v} + -0.25\right)\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;\left(t_0 + -1.5\right) - \frac{r \cdot \left(w \cdot 0.375\right)}{\frac{\frac{1}{w}}{r}}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + t_1 \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* (* r w) (* r w))))
   (if (<= v -480.0)
     (+ -1.5 (+ t_0 (* t_1 (+ (/ 0.125 v) -0.25))))
     (if (<= v 2.5e-99)
       (- (+ t_0 -1.5) (/ (* r (* w 0.375)) (/ (/ 1.0 w) r)))
       (+ -1.5 (+ t_0 (* t_1 -0.25)))))))

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 <= -480.0) {
		tmp = -1.5 + (t_0 + (t_1 * ((0.125 / v) + -0.25)));
	} else if (v <= 2.5e-99) {
		tmp = (t_0 + -1.5) - ((r * (w * 0.375)) / ((1.0 / w) / r));
	} else {
		tmp = -1.5 + (t_0 + (t_1 * -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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (r * w) * (r * w)
    if (v <= (-480.0d0)) then
        tmp = (-1.5d0) + (t_0 + (t_1 * ((0.125d0 / v) + (-0.25d0))))
    else if (v <= 2.5d-99) then
        tmp = (t_0 + (-1.5d0)) - ((r * (w * 0.375d0)) / ((1.0d0 / w) / r))
    else
        tmp = (-1.5d0) + (t_0 + (t_1 * (-0.25d0)))
    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 <= -480.0) {
		tmp = -1.5 + (t_0 + (t_1 * ((0.125 / v) + -0.25)));
	} else if (v <= 2.5e-99) {
		tmp = (t_0 + -1.5) - ((r * (w * 0.375)) / ((1.0 / w) / r));
	} else {
		tmp = -1.5 + (t_0 + (t_1 * -0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (r * w) * (r * w)
	tmp = 0
	if v <= -480.0:
		tmp = -1.5 + (t_0 + (t_1 * ((0.125 / v) + -0.25)))
	elif v <= 2.5e-99:
		tmp = (t_0 + -1.5) - ((r * (w * 0.375)) / ((1.0 / w) / r))
	else:
		tmp = -1.5 + (t_0 + (t_1 * -0.25))
	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 <= -480.0)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(t_1 * Float64(Float64(0.125 / v) + -0.25))));
	elseif (v <= 2.5e-99)
		tmp = Float64(Float64(t_0 + -1.5) - Float64(Float64(r * Float64(w * 0.375)) / Float64(Float64(1.0 / w) / r)));
	else
		tmp = Float64(-1.5 + Float64(t_0 + Float64(t_1 * -0.25)));
	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 <= -480.0)
		tmp = -1.5 + (t_0 + (t_1 * ((0.125 / v) + -0.25)));
	elseif (v <= 2.5e-99)
		tmp = (t_0 + -1.5) - ((r * (w * 0.375)) / ((1.0 / w) / r));
	else
		tmp = -1.5 + (t_0 + (t_1 * -0.25));
	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, -480.0], N[(-1.5 + N[(t$95$0 + N[(t$95$1 * N[(N[(0.125 / v), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2.5e-99], N[(N[(t$95$0 + -1.5), $MachinePrecision] - N[(N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / w), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$0 + N[(t$95$1 * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -480

   1. Initial program 80.6%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in r around 0 86.2%

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

      1. unpow2 86.2%

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

      2. unpow2 86.2%

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

      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 98.6%

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

       1. sub-neg 98.6%

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

       2. associate-*r/ 98.6%

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

       3. metadata-eval 98.6%

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

       4. metadata-eval 98.6%

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

   11. Simplified 98.6%

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

   if -480 < v < 2.49999999999999985e-99

   1. Initial program 92.4%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in v around 0 86.6%

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

      1. *-commutative 86.6%

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

      2. unpow2 86.6%

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

      3. unpow2 86.6%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

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

      1. associate-*r* 99.0%

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

      2. /-rgt-identity 99.0%

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

      3. associate-/r/ 99.0%

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

      4. clear-num 99.0%

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

      5. clear-num 99.0%

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

      6. associate-/l* 99.8%

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

      7. associate-*l/ 99.8%

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

      8. associate-*l* 99.9%

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

   10. Applied egg-rr 99.9%

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

   if 2.49999999999999985e-99 < v

   1. Initial program 81.9%

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

   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 81.5%

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

      1. unpow2 81.5%

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

      2. unpow2 81.5%

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

      3. swap-sqr 99.9%

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

      4. unpow2 99.9%

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

   6. Simplified 99.9%

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

      1. unpow2 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in v around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -480:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(\frac{0.125}{v} + -0.25\right)\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(w \cdot 0.375\right)}{\frac{\frac{1}{w}}{r}}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -470 \lor \neg \left(v \leq 2.5 \cdot 10^{-99}\right):\\ \;\;\;\;-1.5 + \left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + -1.5\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -470.0) (not (<= v 2.5e-99)))
     (+ -1.5 (+ t_0 (* (* (* r w) (* r w)) -0.25)))
     (- (+ t_0 -1.5) (* (* r w) (* r (* w 0.375)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -470.0) || !(v <= 2.5e-99)) {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	} else {
		tmp = (t_0 + -1.5) - ((r * w) * (r * (w * 0.375)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-470.0d0)) .or. (.not. (v <= 2.5d-99))) then
        tmp = (-1.5d0) + (t_0 + (((r * w) * (r * w)) * (-0.25d0)))
    else
        tmp = (t_0 + (-1.5d0)) - ((r * w) * (r * (w * 0.375d0)))
    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 <= -470.0) || !(v <= 2.5e-99)) {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	} else {
		tmp = (t_0 + -1.5) - ((r * w) * (r * (w * 0.375)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -470.0) or not (v <= 2.5e-99):
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25))
	else:
		tmp = (t_0 + -1.5) - ((r * w) * (r * (w * 0.375)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -470.0) || !(v <= 2.5e-99))
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25)));
	else
		tmp = Float64(Float64(t_0 + -1.5) - Float64(Float64(r * w) * Float64(r * Float64(w * 0.375))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -470.0) || ~((v <= 2.5e-99)))
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	else
		tmp = (t_0 + -1.5) - ((r * w) * (r * (w * 0.375)));
	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, -470.0], N[Not[LessEqual[v, 2.5e-99]], $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[(N[(t$95$0 + -1.5), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -470 or 2.49999999999999985e-99 < v

   1. Initial program 81.3%

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

   3. Simplified 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 83.5%

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

      1. unpow2 83.5%

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

      2. unpow2 83.5%

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

      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 inf 99.0%

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

   if -470 < v < 2.49999999999999985e-99

   1. Initial program 92.4%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in v around 0 86.6%

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

      1. *-commutative 86.6%

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

      2. unpow2 86.6%

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

      3. unpow2 86.6%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

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

      1. associate-*r* 99.0%

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

      2. /-rgt-identity 99.0%

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

      3. associate-/r/ 99.0%

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

      4. clear-num 99.0%

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

      5. clear-num 99.0%

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

      6. associate-/l* 99.8%

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

      7. associate-*l/ 99.8%

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

      8. associate-*l* 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. div-inv 99.9%

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

       2. associate-/l/ 99.9%

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

       3. remove-double-div 99.9%

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

   12. Applied egg-rr 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -470 \lor \neg \left(v \leq 2.5 \cdot 10^{-99}\right):\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\\ \end{array} \]

Alternative 5: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -500:\\ \;\;\;\;-1.5 + \left(t_0 + -0.25 \cdot \frac{w}{\frac{\frac{\frac{1}{w}}{r}}{r}}\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;\left(t_0 + -1.5\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -500.0)
     (+ -1.5 (+ t_0 (* -0.25 (/ w (/ (/ (/ 1.0 w) r) r)))))
     (if (<= v 2.5e-99)
       (- (+ t_0 -1.5) (* (* r w) (* r (* w 0.375))))
       (+ -1.5 (+ t_0 (* (* (* r w) (* r w)) -0.25)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -500.0) {
		tmp = -1.5 + (t_0 + (-0.25 * (w / (((1.0 / w) / r) / r))));
	} else if (v <= 2.5e-99) {
		tmp = (t_0 + -1.5) - ((r * w) * (r * (w * 0.375)));
	} else {
		tmp = -1.5 + (t_0 + (((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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (v <= (-500.0d0)) then
        tmp = (-1.5d0) + (t_0 + ((-0.25d0) * (w / (((1.0d0 / w) / r) / r))))
    else if (v <= 2.5d-99) then
        tmp = (t_0 + (-1.5d0)) - ((r * w) * (r * (w * 0.375d0)))
    else
        tmp = (-1.5d0) + (t_0 + (((r * w) * (r * w)) * (-0.25d0)))
    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 <= -500.0) {
		tmp = -1.5 + (t_0 + (-0.25 * (w / (((1.0 / w) / r) / r))));
	} else if (v <= 2.5e-99) {
		tmp = (t_0 + -1.5) - ((r * w) * (r * (w * 0.375)));
	} else {
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -500.0:
		tmp = -1.5 + (t_0 + (-0.25 * (w / (((1.0 / w) / r) / r))))
	elif v <= 2.5e-99:
		tmp = (t_0 + -1.5) - ((r * w) * (r * (w * 0.375)))
	else:
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -500.0)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(-0.25 * Float64(w / Float64(Float64(Float64(1.0 / w) / r) / r)))));
	elseif (v <= 2.5e-99)
		tmp = Float64(Float64(t_0 + -1.5) - Float64(Float64(r * w) * Float64(r * Float64(w * 0.375))));
	else
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -500.0)
		tmp = -1.5 + (t_0 + (-0.25 * (w / (((1.0 / w) / r) / r))));
	elseif (v <= 2.5e-99)
		tmp = (t_0 + -1.5) - ((r * w) * (r * (w * 0.375)));
	else
		tmp = -1.5 + (t_0 + (((r * w) * (r * w)) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -500.0], N[(-1.5 + N[(t$95$0 + N[(-0.25 * N[(w / N[(N[(N[(1.0 / w), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2.5e-99], N[(N[(t$95$0 + -1.5), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$0 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -500

   1. Initial program 80.6%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in r around 0 86.2%

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

      1. unpow2 86.2%

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

      2. unpow2 86.2%

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

      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 97.9%

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

       1. associate-*r* 98.0%

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

       2. /-rgt-identity 98.0%

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

       3. associate-/r/ 98.0%

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

       4. associate-/l* 97.9%

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

       5. *-commutative 97.9%

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

       6. associate-/l* 98.0%

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

   11. Applied egg-rr 98.0%

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

   if -500 < v < 2.49999999999999985e-99

   1. Initial program 92.4%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in v around 0 86.6%

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

      1. *-commutative 86.6%

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

      2. unpow2 86.6%

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

      3. unpow2 86.6%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

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

      1. associate-*r* 99.0%

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

      2. /-rgt-identity 99.0%

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

      3. associate-/r/ 99.0%

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

      4. clear-num 99.0%

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

      5. clear-num 99.0%

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

      6. associate-/l* 99.8%

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

      7. associate-*l/ 99.8%

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

      8. associate-*l* 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. div-inv 99.9%

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

       2. associate-/l/ 99.9%

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

       3. remove-double-div 99.9%

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

   12. Applied egg-rr 99.9%

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

   if 2.49999999999999985e-99 < v

   1. Initial program 81.9%

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

   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 81.5%

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

      1. unpow2 81.5%

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

      2. unpow2 81.5%

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

      3. swap-sqr 99.9%

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

      4. unpow2 99.9%

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

   6. Simplified 99.9%

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

      1. unpow2 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in v around inf 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -500:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.25 \cdot \frac{w}{\frac{\frac{\frac{1}{w}}{r}}{r}}\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 6: 93.0% 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 86.1%

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

3. Simplified 89.4%

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

4. Taylor expanded in r around 0 84.8%

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

   1. unpow2 84.8%

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

   2. unpow2 84.8%

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

   3. swap-sqr 99.8%

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

   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 95.7%

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

10. Final simplification 95.7%

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

Reproduce

herbie shell --seed 2023318 
(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))