Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.5%

Time: 12.8s

Alternatives: 17

Speedup: 1.7×

• 54.2% 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: 85.0% 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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(r \cdot w\right)}^{2}\\ t_1 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot t_0\right)\\ \mathbf{elif}\;v \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;t_1 - \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{\frac{1}{r}}{w}}\\ \mathbf{else}:\\ \;\;\;\;t_1 - t_0 \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (pow (* r w) 2.0)) (t_1 (+ (/ 2.0 (* r r)) -1.5)))
   (if (<= v -5.5e-5)
     (+ -1.5 (+ (/ (/ 2.0 r) r) (* -0.25 t_0)))
     (if (<= v 2.6e-9)
       (- t_1 (/ (* (* r w) 0.375) (/ (/ 1.0 r) w)))
       (- t_1 (* t_0 0.25))))))

C

double code(double v, double w, double r) {
	double t_0 = pow((r * w), 2.0);
	double t_1 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (v <= -5.5e-5) {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * t_0));
	} else if (v <= 2.6e-9) {
		tmp = t_1 - (((r * w) * 0.375) / ((1.0 / r) / w));
	} else {
		tmp = t_1 - (t_0 * 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 = (r * w) ** 2.0d0
    t_1 = (2.0d0 / (r * r)) + (-1.5d0)
    if (v <= (-5.5d-5)) then
        tmp = (-1.5d0) + (((2.0d0 / r) / r) + ((-0.25d0) * t_0))
    else if (v <= 2.6d-9) then
        tmp = t_1 - (((r * w) * 0.375d0) / ((1.0d0 / r) / w))
    else
        tmp = t_1 - (t_0 * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = Math.pow((r * w), 2.0);
	double t_1 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (v <= -5.5e-5) {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * t_0));
	} else if (v <= 2.6e-9) {
		tmp = t_1 - (((r * w) * 0.375) / ((1.0 / r) / w));
	} else {
		tmp = t_1 - (t_0 * 0.25);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = math.pow((r * w), 2.0)
	t_1 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if v <= -5.5e-5:
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * t_0))
	elif v <= 2.6e-9:
		tmp = t_1 - (((r * w) * 0.375) / ((1.0 / r) / w))
	else:
		tmp = t_1 - (t_0 * 0.25)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * w) ^ 2.0
	t_1 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if (v <= -5.5e-5)
		tmp = Float64(-1.5 + Float64(Float64(Float64(2.0 / r) / r) + Float64(-0.25 * t_0)));
	elseif (v <= 2.6e-9)
		tmp = Float64(t_1 - Float64(Float64(Float64(r * w) * 0.375) / Float64(Float64(1.0 / r) / w)));
	else
		tmp = Float64(t_1 - Float64(t_0 * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) ^ 2.0;
	t_1 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if (v <= -5.5e-5)
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * t_0));
	elseif (v <= 2.6e-9)
		tmp = t_1 - (((r * w) * 0.375) / ((1.0 / r) / w));
	else
		tmp = t_1 - (t_0 * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[LessEqual[v, -5.5e-5], N[(-1.5 + N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-0.25 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2.6e-9], N[(t$95$1 - N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.5000000000000002e-5

   1. Initial program 83.9%

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

   3. Simplified 86.9%

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

      1. associate-/r* 87.0%

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

      2. div-inv 86.9%

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

   5. Applied egg-rr 86.9%

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

      1. un-div-inv 87.0%

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

   7. Applied egg-rr 87.0%

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

   8. Taylor expanded in v around inf 83.6%

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

      1. *-commutative 83.6%

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

      2. unpow2 83.6%

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

      3. unpow2 83.6%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   10. Simplified 99.8%

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

   if -5.5000000000000002e-5 < v < 2.6000000000000001e-9

   1. Initial program 90.6%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in v around 0 83.9%

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

      1. *-commutative 83.9%

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

      2. unpow2 83.9%

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

      3. unpow2 83.9%

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

      4. swap-sqr 99.7%

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

      5. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. pow2 99.7%

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

      3. metadata-eval 99.7%

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

      4. pow-div 99.7%

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

      5. pow1 99.7%

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

      6. inv-pow 99.7%

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

      7. associate-*r/ 99.8%

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

      8. associate-/r* 99.8%

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

   10. Applied egg-rr 99.8%

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

   if 2.6000000000000001e-9 < v

   1. Initial program 78.0%

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

   3. Simplified 96.2%

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

      1. associate-*l/ 86.9%

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

      2. associate-/l* 96.3%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

      2. /-rgt-identity 99.8%

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

      3. /-rgt-identity 99.8%

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

      4. associate-/l* 99.9%

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

      5. frac-times 95.2%

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

      6. *-un-lft-identity 95.2%

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

   7. Applied egg-rr 95.2%

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

   8. Taylor expanded in v around inf 72.6%

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

      1. unpow2 72.6%

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

      2. unpow2 72.6%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{elif}\;v \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{\frac{1}{r}}{w}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - {\left(r \cdot w\right)}^{2} \cdot 0.25\\ \end{array} \]

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{1 - v}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= (* w w) 2e+270)
   (-
    (+ (/ 2.0 (* r r)) -1.5)
    (* (fma v -0.25 0.375) (/ (* r (* w (* r w))) (- 1.0 v))))
   (+ -1.5 (+ (/ (/ 2.0 r) r) (* -0.25 (pow (* r w) 2.0))))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((w * w) <= 2e+270) {
		tmp = ((2.0 / (r * r)) + -1.5) - (fma(v, -0.25, 0.375) * ((r * (w * (r * w))) / (1.0 - v)));
	} else {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * pow((r * w), 2.0)));
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (Float64(w * w) <= 2e+270)
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + -1.5) - Float64(fma(v, -0.25, 0.375) * Float64(Float64(r * Float64(w * Float64(r * w))) / Float64(1.0 - v))));
	else
		tmp = Float64(-1.5 + Float64(Float64(Float64(2.0 / r) / r) + Float64(-0.25 * (Float64(r * w) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(w * w), $MachinePrecision], 2e+270], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] - N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-0.25 * N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 2.0000000000000001e270

   1. Initial program 93.4%

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

   3. Simplified 99.7%

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

   if 2.0000000000000001e270 < (*.f64 w w)

   1. Initial program 66.9%

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

   3. Simplified 69.2%

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

      1. associate-/r* 69.2%

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

      2. div-inv 69.2%

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

   5. Applied egg-rr 69.2%

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

      1. un-div-inv 69.2%

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

   7. Applied egg-rr 69.2%

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

   8. Taylor expanded in v around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. unpow2 69.2%

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

      3. unpow2 69.2%

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

      4. swap-sqr 99.9%

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

      5. unpow2 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{1 - v}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \end{array} \]

Alternative 3: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-56}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - r \cdot \left(w \cdot \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot \frac{w}{1 - v}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4e-56)
   (+ -1.5 (+ (/ (/ 2.0 r) r) (* -0.25 (pow (* r w) 2.0))))
   (-
    (+ (/ 2.0 (* r r)) -1.5)
    (* r (* w (* (fma v -0.25 0.375) (* r (/ w (- 1.0 v)))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4e-56) {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * pow((r * w), 2.0)));
	} else {
		tmp = ((2.0 / (r * r)) + -1.5) - (r * (w * (fma(v, -0.25, 0.375) * (r * (w / (1.0 - v))))));
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 4e-56)
		tmp = Float64(-1.5 + Float64(Float64(Float64(2.0 / r) / r) + Float64(-0.25 * (Float64(r * w) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + -1.5) - Float64(r * Float64(w * Float64(fma(v, -0.25, 0.375) * Float64(r * Float64(w / Float64(1.0 - v)))))));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4e-56], N[(-1.5 + N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-0.25 * N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] - N[(r * N[(w * N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(r * N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.0000000000000002e-56

   1. Initial program 82.5%

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

   3. Simplified 84.0%

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

      1. associate-/r* 84.1%

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

      2. div-inv 84.0%

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

   5. Applied egg-rr 84.0%

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

      1. un-div-inv 84.1%

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

   7. Applied egg-rr 84.1%

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

   8. Taylor expanded in v around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. unpow2 78.1%

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

      4. swap-sqr 95.6%

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

      5. unpow2 95.6%

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

   10. Simplified 95.6%

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

   if 4.0000000000000002e-56 < r

   1. Initial program 91.9%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 93.3%

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

      2. associate-/l* 99.7%

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

      3. associate-*r* 99.7%

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

      4. pow2 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r/ 99.7%

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

      2. unpow2 99.7%

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

      3. associate-*r/ 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-*l* 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. times-frac 99.3%

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

      8. /-rgt-identity 99.3%

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-56}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - r \cdot \left(w \cdot \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot \frac{w}{1 - v}\right)\right)\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Simplified 96.8%

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

   1. associate-*r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 5: 96.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 85000:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 - \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 85000.0)
   (+ -1.5 (+ (/ (/ 2.0 r) r) (* -0.25 (pow (* r w) 2.0))))
   (- -1.5 (* (* (* r w) (/ (* r w) (- 1.0 v))) (fma v -0.25 0.375)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 85000.0) {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * pow((r * w), 2.0)));
	} else {
		tmp = -1.5 - (((r * w) * ((r * w) / (1.0 - v))) * fma(v, -0.25, 0.375));
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 85000.0)
		tmp = Float64(-1.5 + Float64(Float64(Float64(2.0 / r) / r) + Float64(-0.25 * (Float64(r * w) ^ 2.0))));
	else
		tmp = Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))) * fma(v, -0.25, 0.375)));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 85000.0], N[(-1.5 + N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-0.25 * N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 85000

   1. Initial program 83.1%

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

   3. Simplified 85.0%

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

      1. associate-/r* 85.0%

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

      2. div-inv 84.9%

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

   5. Applied egg-rr 84.9%

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

      1. un-div-inv 85.0%

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

   7. Applied egg-rr 85.0%

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

   8. Taylor expanded in v around inf 78.6%

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

      1. *-commutative 78.6%

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

      2. unpow2 78.6%

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

      3. unpow2 78.6%

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

      4. swap-sqr 95.0%

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

      5. unpow2 95.0%

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

   10. Simplified 95.0%

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

   if 85000 < r

   1. Initial program 92.0%

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

   3. Simplified 99.8%

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

      1. associate-*r* 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. times-frac 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in r around inf 99.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 85000:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 - \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\\ \end{array} \]

Alternative 6: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.5 \cdot 10^{-5} \lor \neg \left(v \leq 5 \cdot 10^{+29}\right):\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{\frac{1}{r}}{w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -5.5e-5) (not (<= v 5e+29)))
   (+ -1.5 (+ (/ (/ 2.0 r) r) (* -0.25 (pow (* r w) 2.0))))
   (- (+ (/ 2.0 (* r r)) -1.5) (/ (* (* r w) 0.375) (/ (/ 1.0 r) w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -5.5e-5) || !(v <= 5e+29)) {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * pow((r * w), 2.0)));
	} else {
		tmp = ((2.0 / (r * r)) + -1.5) - (((r * w) * 0.375) / ((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) :: tmp
    if ((v <= (-5.5d-5)) .or. (.not. (v <= 5d+29))) then
        tmp = (-1.5d0) + (((2.0d0 / r) / r) + ((-0.25d0) * ((r * w) ** 2.0d0)))
    else
        tmp = ((2.0d0 / (r * r)) + (-1.5d0)) - (((r * w) * 0.375d0) / ((1.0d0 / r) / w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -5.5e-5) || !(v <= 5e+29)) {
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * Math.pow((r * w), 2.0)));
	} else {
		tmp = ((2.0 / (r * r)) + -1.5) - (((r * w) * 0.375) / ((1.0 / r) / w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -5.5e-5) or not (v <= 5e+29):
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * math.pow((r * w), 2.0)))
	else:
		tmp = ((2.0 / (r * r)) + -1.5) - (((r * w) * 0.375) / ((1.0 / r) / w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -5.5e-5) || !(v <= 5e+29))
		tmp = Float64(-1.5 + Float64(Float64(Float64(2.0 / r) / r) + Float64(-0.25 * (Float64(r * w) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + -1.5) - Float64(Float64(Float64(r * w) * 0.375) / Float64(Float64(1.0 / r) / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -5.5e-5) || ~((v <= 5e+29)))
		tmp = -1.5 + (((2.0 / r) / r) + (-0.25 * ((r * w) ^ 2.0)));
	else
		tmp = ((2.0 / (r * r)) + -1.5) - (((r * w) * 0.375) / ((1.0 / r) / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -5.5e-5], N[Not[LessEqual[v, 5e+29]], $MachinePrecision]], N[(-1.5 + N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-0.25 * N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] - N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -5.5000000000000002e-5 or 5.0000000000000001e29 < v

   1. Initial program 80.0%

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

   3. Simplified 85.2%

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

      1. associate-/r* 85.2%

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

      2. div-inv 85.1%

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

   5. Applied egg-rr 85.1%

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

      1. un-div-inv 85.2%

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

   7. Applied egg-rr 85.2%

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

   8. Taylor expanded in v around inf 76.6%

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

      1. *-commutative 76.6%

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

      2. unpow2 76.6%

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

      3. unpow2 76.6%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   10. Simplified 99.8%

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

   if -5.5000000000000002e-5 < v < 5.0000000000000001e29

   1. Initial program 90.5%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in v around 0 84.3%

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

      1. *-commutative 84.3%

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

      2. unpow2 84.3%

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

      3. unpow2 84.3%

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

      4. swap-sqr 99.7%

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

      5. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. pow2 99.7%

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

      3. metadata-eval 99.7%

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

      4. pow-div 99.7%

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

      5. pow1 99.7%

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

      6. inv-pow 99.7%

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

      7. associate-*r/ 99.8%

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

      8. associate-/r* 99.8%

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

   10. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5.5 \cdot 10^{-5} \lor \neg \left(v \leq 5 \cdot 10^{+29}\right):\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{\frac{1}{r}}{w}}\\ \end{array} \]

Alternative 7: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{+306}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 + \frac{\frac{2}{r}}{r}\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= (* w w) 4e+306)
   (+
    -1.5
    (+
     (/ 2.0 (* r r))
     (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r (* r (* w w))))))
   (- (+ -1.5 (/ (/ 2.0 r) r)) (* 0.375 (* (* r w) (* r w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((w * w) <= 4e+306) {
		tmp = -1.5 + ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = (-1.5 + ((2.0 / r) / r)) - (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) :: tmp
    if ((w * w) <= 4d+306) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r * (r * (w * w)))))
    else
        tmp = ((-1.5d0) + ((2.0d0 / r) / r)) - (0.375d0 * ((r * w) * (r * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((w * w) <= 4e+306) {
		tmp = -1.5 + ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = (-1.5 + ((2.0 / r) / r)) - (0.375 * ((r * w) * (r * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (w * w) <= 4e+306:
		tmp = -1.5 + ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	else:
		tmp = (-1.5 + ((2.0 / r) / r)) - (0.375 * ((r * w) * (r * w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (Float64(w * w) <= 4e+306)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))));
	else
		tmp = Float64(Float64(-1.5 + Float64(Float64(2.0 / r) / r)) - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((w * w) <= 4e+306)
		tmp = -1.5 + ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	else
		tmp = (-1.5 + ((2.0 / r) / r)) - (0.375 * ((r * w) * (r * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(w * w), $MachinePrecision], 4e+306], N[(-1.5 + 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[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 4.00000000000000007e306

   1. Initial program 92.7%

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

   3. Simplified 96.4%

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

   if 4.00000000000000007e306 < (*.f64 w w)

   1. Initial program 66.7%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in v around 0 66.7%

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

      1. *-commutative 66.7%

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

      2. unpow2 66.7%

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

      3. unpow2 66.7%

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

      4. swap-sqr 95.7%

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

      5. unpow2 95.7%

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

   6. Simplified 95.7%

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

      1. unpow2 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. clear-num 95.7%

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

      2. associate-/r/ 95.7%

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

      3. pow2 95.7%

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

      4. pow-flip 95.7%

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

      5. metadata-eval 95.7%

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

   10. Applied egg-rr 95.7%

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

       1. *-commutative 95.7%

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

       2. metadata-eval 95.7%

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

       3. pow-prod-up 95.7%

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

       4. inv-pow 95.7%

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

       5. inv-pow 95.7%

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

       6. associate-*r* 95.7%

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

       7. div-inv 95.7%

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

       8. un-div-inv 95.7%

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

   12. Applied egg-rr 95.7%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{+306}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 + \frac{\frac{2}{r}}{r}\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 8: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{+306}:\\ \;\;\;\;-1.5 + \left(t_0 + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 + t_0\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (<= (* w w) 4e+306)
     (+
      -1.5
      (+ t_0 (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r (* r (* w w))))))
     (- (+ -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 ((w * w) <= 4e+306) {
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} 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 ((w * w) <= 4d+306) then
        tmp = (-1.5d0) + (t_0 + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r * (r * (w * w)))))
    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 ((w * w) <= 4e+306) {
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} 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 (w * w) <= 4e+306:
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	else:
		tmp = (-1.5 + t_0) - (0.375 * ((r * w) * (r * w)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if (Float64(w * w) <= 4e+306)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))));
	else
		tmp = Float64(Float64(-1.5 + t_0) - Float64(0.375 * Float64(Float64(r * 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 ((w * w) <= 4e+306)
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	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[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 4e+306], N[(-1.5 + N[(t$95$0 + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.5 + t$95$0), $MachinePrecision] - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 4.00000000000000007e306

   1. Initial program 92.7%

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

   3. Simplified 96.4%

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

      1. associate-/r* 96.4%

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

      2. div-inv 96.3%

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

   5. Applied egg-rr 96.3%

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

      1. un-div-inv 96.4%

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

   7. Applied egg-rr 96.4%

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

   if 4.00000000000000007e306 < (*.f64 w w)

   1. Initial program 66.7%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in v around 0 66.7%

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

      1. *-commutative 66.7%

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

      2. unpow2 66.7%

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

      3. unpow2 66.7%

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

      4. swap-sqr 95.7%

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

      5. unpow2 95.7%

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

   6. Simplified 95.7%

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

      1. unpow2 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. clear-num 95.7%

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

      2. associate-/r/ 95.7%

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

      3. pow2 95.7%

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

      4. pow-flip 95.7%

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

      5. metadata-eval 95.7%

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

   10. Applied egg-rr 95.7%

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

       1. *-commutative 95.7%

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

       2. metadata-eval 95.7%

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

       3. pow-prod-up 95.7%

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

       4. inv-pow 95.7%

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

       5. inv-pow 95.7%

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

       6. associate-*r* 95.7%

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

       7. div-inv 95.7%

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

       8. un-div-inv 95.7%

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

   12. Applied egg-rr 95.7%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{+306}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 + \frac{\frac{2}{r}}{r}\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 9: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq -14200000000 \lor \neg \left(v \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;t_0 - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{\frac{1}{r}}{w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5)))
   (if (or (<= v -14200000000.0) (not (<= v 2e-12)))
     (- t_0 (* r (* w (* w (* r 0.25)))))
     (- t_0 (/ (* (* r w) 0.375) (/ (/ 1.0 r) w))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if ((v <= -14200000000.0) || !(v <= 2e-12)) {
		tmp = t_0 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = t_0 - (((r * w) * 0.375) / ((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)) + (-1.5d0)
    if ((v <= (-14200000000.0d0)) .or. (.not. (v <= 2d-12))) then
        tmp = t_0 - (r * (w * (w * (r * 0.25d0))))
    else
        tmp = t_0 - (((r * w) * 0.375d0) / ((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)) + -1.5;
	double tmp;
	if ((v <= -14200000000.0) || !(v <= 2e-12)) {
		tmp = t_0 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = t_0 - (((r * w) * 0.375) / ((1.0 / r) / w));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if (v <= -14200000000.0) or not (v <= 2e-12):
		tmp = t_0 - (r * (w * (w * (r * 0.25))))
	else:
		tmp = t_0 - (((r * w) * 0.375) / ((1.0 / r) / w))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if ((v <= -14200000000.0) || !(v <= 2e-12))
		tmp = Float64(t_0 - Float64(r * Float64(w * Float64(w * Float64(r * 0.25)))));
	else
		tmp = Float64(t_0 - Float64(Float64(Float64(r * w) * 0.375) / Float64(Float64(1.0 / r) / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if ((v <= -14200000000.0) || ~((v <= 2e-12)))
		tmp = t_0 - (r * (w * (w * (r * 0.25))));
	else
		tmp = t_0 - (((r * w) * 0.375) / ((1.0 / r) / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[Or[LessEqual[v, -14200000000.0], N[Not[LessEqual[v, 2e-12]], $MachinePrecision]], N[(t$95$0 - N[(r * N[(w * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.42e10 or 1.99999999999999996e-12 < v

   1. Initial program 80.0%

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

   3. Simplified 95.5%

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

      1. associate-*l/ 88.0%

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

      2. associate-/l* 95.6%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-*l* 95.6%

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

      6. *-un-lft-identity 95.6%

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

      7. times-frac 93.9%

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

      8. /-rgt-identity 93.9%

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

   7. Applied egg-rr 93.9%

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

   8. Taylor expanded in v around inf 95.6%

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

      1. associate-*r* 95.6%

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

   10. Simplified 95.6%

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

   if -1.42e10 < v < 1.99999999999999996e-12

   1. Initial program 90.9%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in v around 0 84.4%

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

      1. *-commutative 84.4%

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

      2. unpow2 84.4%

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

      3. unpow2 84.4%

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

      4. swap-sqr 99.7%

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

      5. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. pow2 99.7%

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

      3. metadata-eval 99.7%

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

      4. pow-div 99.7%

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

      5. pow1 99.7%

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

      6. inv-pow 99.7%

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

      7. associate-*r/ 99.7%

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

      8. associate-/r* 99.8%

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

   10. Applied egg-rr 99.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -14200000000 \lor \neg \left(v \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{\frac{1}{r}}{w}}\\ \end{array} \]

Alternative 10: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq -2100000000 \lor \neg \left(v \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;t_0 - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot 0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5)))
   (if (or (<= v -2100000000.0) (not (<= v 2.6e-9)))
     (- t_0 (* r (* w (* w (* r 0.25)))))
     (- t_0 (* r (* w (* (* r w) 0.375)))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if ((v <= -2100000000.0) || !(v <= 2.6e-9)) {
		tmp = t_0 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = t_0 - (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)) + (-1.5d0)
    if ((v <= (-2100000000.0d0)) .or. (.not. (v <= 2.6d-9))) then
        tmp = t_0 - (r * (w * (w * (r * 0.25d0))))
    else
        tmp = t_0 - (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)) + -1.5;
	double tmp;
	if ((v <= -2100000000.0) || !(v <= 2.6e-9)) {
		tmp = t_0 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = t_0 - (r * (w * ((r * w) * 0.375)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if (v <= -2100000000.0) or not (v <= 2.6e-9):
		tmp = t_0 - (r * (w * (w * (r * 0.25))))
	else:
		tmp = t_0 - (r * (w * ((r * w) * 0.375)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if ((v <= -2100000000.0) || !(v <= 2.6e-9))
		tmp = Float64(t_0 - Float64(r * Float64(w * Float64(w * Float64(r * 0.25)))));
	else
		tmp = Float64(t_0 - Float64(r * Float64(w * Float64(Float64(r * w) * 0.375))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if ((v <= -2100000000.0) || ~((v <= 2.6e-9)))
		tmp = t_0 - (r * (w * (w * (r * 0.25))));
	else
		tmp = t_0 - (r * (w * ((r * w) * 0.375)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[Or[LessEqual[v, -2100000000.0], N[Not[LessEqual[v, 2.6e-9]], $MachinePrecision]], N[(t$95$0 - N[(r * N[(w * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(r * N[(w * N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.1e9 or 2.6000000000000001e-9 < v

   1. Initial program 80.0%

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

   3. Simplified 95.5%

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

      1. associate-*l/ 88.0%

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

      2. associate-/l* 95.6%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-*l* 95.6%

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

      6. *-un-lft-identity 95.6%

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

      7. times-frac 93.9%

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

      8. /-rgt-identity 93.9%

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

   7. Applied egg-rr 93.9%

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

   8. Taylor expanded in v around inf 95.6%

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

      1. associate-*r* 95.6%

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

   10. Simplified 95.6%

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

   if -2.1e9 < v < 2.6000000000000001e-9

   1. Initial program 90.9%

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

   3. Simplified 98.1%

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

      1. associate-*l/ 98.1%

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

      2. associate-/l* 98.1%

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

      3. associate-*r* 99.7%

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

      4. pow2 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-/r/ 99.7%

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

      2. unpow2 99.7%

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

      3. associate-*r/ 99.7%

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

      4. associate-*l* 99.8%

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

      5. associate-*l* 98.1%

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

      6. *-un-lft-identity 98.1%

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

      7. times-frac 98.1%

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

      8. /-rgt-identity 98.1%

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

   7. Applied egg-rr 98.1%

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

   8. Taylor expanded in v around 0 98.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2100000000 \lor \neg \left(v \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot 0.375\right)\right)\\ \end{array} \]

Alternative 11: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq -5 \cdot 10^{+15} \lor \neg \left(v \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;t_0 - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5)))
   (if (or (<= v -5e+15) (not (<= v 2.6e-9)))
     (- t_0 (* r (* w (* w (* r 0.25)))))
     (- t_0 (* 0.375 (* (* r w) (* r w)))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if ((v <= -5e+15) || !(v <= 2.6e-9)) {
		tmp = t_0 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = 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)) + (-1.5d0)
    if ((v <= (-5d+15)) .or. (.not. (v <= 2.6d-9))) then
        tmp = t_0 - (r * (w * (w * (r * 0.25d0))))
    else
        tmp = 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)) + -1.5;
	double tmp;
	if ((v <= -5e+15) || !(v <= 2.6e-9)) {
		tmp = t_0 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = t_0 - (0.375 * ((r * w) * (r * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if (v <= -5e+15) or not (v <= 2.6e-9):
		tmp = t_0 - (r * (w * (w * (r * 0.25))))
	else:
		tmp = t_0 - (0.375 * ((r * w) * (r * w)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if ((v <= -5e+15) || !(v <= 2.6e-9))
		tmp = Float64(t_0 - Float64(r * Float64(w * Float64(w * Float64(r * 0.25)))));
	else
		tmp = Float64(t_0 - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if ((v <= -5e+15) || ~((v <= 2.6e-9)))
		tmp = t_0 - (r * (w * (w * (r * 0.25))));
	else
		tmp = t_0 - (0.375 * ((r * w) * (r * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[Or[LessEqual[v, -5e+15], N[Not[LessEqual[v, 2.6e-9]], $MachinePrecision]], N[(t$95$0 - N[(r * N[(w * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -5e15 or 2.6000000000000001e-9 < v

   1. Initial program 80.0%

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

   3. Simplified 95.5%

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

      1. associate-*l/ 88.0%

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

      2. associate-/l* 95.6%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-*l* 95.6%

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

      6. *-un-lft-identity 95.6%

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

      7. times-frac 93.9%

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

      8. /-rgt-identity 93.9%

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

   7. Applied egg-rr 93.9%

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

   8. Taylor expanded in v around inf 95.6%

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

      1. associate-*r* 95.6%

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

   10. Simplified 95.6%

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

   if -5e15 < v < 2.6000000000000001e-9

   1. Initial program 90.9%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in v around 0 84.4%

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

      1. *-commutative 84.4%

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

      2. unpow2 84.4%

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

      3. unpow2 84.4%

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

      4. swap-sqr 99.7%

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

      5. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+15} \lor \neg \left(v \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 12: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2100000000 \lor \neg \left(v \leq 0.00033\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 + \frac{\frac{2}{r}}{r}\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -2100000000.0) (not (<= v 0.00033)))
   (- (+ (/ 2.0 (* r r)) -1.5) (* r (* w (* w (* r 0.25)))))
   (- (+ -1.5 (/ (/ 2.0 r) r)) (* 0.375 (* (* r w) (* r w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -2100000000.0) || !(v <= 0.00033)) {
		tmp = ((2.0 / (r * r)) + -1.5) - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = (-1.5 + ((2.0 / r) / r)) - (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) :: tmp
    if ((v <= (-2100000000.0d0)) .or. (.not. (v <= 0.00033d0))) then
        tmp = ((2.0d0 / (r * r)) + (-1.5d0)) - (r * (w * (w * (r * 0.25d0))))
    else
        tmp = ((-1.5d0) + ((2.0d0 / r) / r)) - (0.375d0 * ((r * w) * (r * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -2100000000.0) || !(v <= 0.00033)) {
		tmp = ((2.0 / (r * r)) + -1.5) - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = (-1.5 + ((2.0 / r) / r)) - (0.375 * ((r * w) * (r * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -2100000000.0) or not (v <= 0.00033):
		tmp = ((2.0 / (r * r)) + -1.5) - (r * (w * (w * (r * 0.25))))
	else:
		tmp = (-1.5 + ((2.0 / r) / r)) - (0.375 * ((r * w) * (r * w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -2100000000.0) || !(v <= 0.00033))
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + -1.5) - Float64(r * Float64(w * Float64(w * Float64(r * 0.25)))));
	else
		tmp = Float64(Float64(-1.5 + Float64(Float64(2.0 / r) / r)) - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -2100000000.0) || ~((v <= 0.00033)))
		tmp = ((2.0 / (r * r)) + -1.5) - (r * (w * (w * (r * 0.25))));
	else
		tmp = (-1.5 + ((2.0 / r) / r)) - (0.375 * ((r * w) * (r * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -2100000000.0], N[Not[LessEqual[v, 0.00033]], $MachinePrecision]], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] - N[(r * N[(w * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.1e9 or 3.3e-4 < v

   1. Initial program 79.7%

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

   3. Simplified 95.5%

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

      1. associate-*l/ 87.9%

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

      2. associate-/l* 95.5%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-*l* 95.5%

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

      6. *-un-lft-identity 95.5%

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

      7. times-frac 93.8%

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

      8. /-rgt-identity 93.8%

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

   7. Applied egg-rr 93.8%

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

   8. Taylor expanded in v around inf 95.5%

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

      1. associate-*r* 95.5%

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

   10. Simplified 95.5%

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

   if -2.1e9 < v < 3.3e-4

   1. Initial program 91.0%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in v around 0 84.7%

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

      1. *-commutative 84.7%

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

      2. unpow2 84.7%

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

      3. unpow2 84.7%

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

      4. swap-sqr 99.7%

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

      5. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. pow2 99.7%

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

      4. pow-flip 99.8%

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

      5. metadata-eval 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. *-commutative 99.8%

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

       2. metadata-eval 99.8%

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

       3. pow-prod-up 99.7%

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

       4. inv-pow 99.7%

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

       5. inv-pow 99.7%

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

       6. associate-*r* 99.7%

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

       7. div-inv 99.7%

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

       8. un-div-inv 99.7%

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

   12. Applied egg-rr 99.7%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2100000000 \lor \neg \left(v \leq 0.00033\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 + \frac{\frac{2}{r}}{r}\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 13: 55.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2200000000 \lor \neg \left(v \leq 2.4 \cdot 10^{-9}\right):\\ \;\;\;\;-1.5 - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 - 0.375 \cdot \frac{r \cdot w}{\frac{\frac{1}{r}}{w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -2200000000.0) (not (<= v 2.4e-9)))
   (- -1.5 (* r (* w (* w (* r 0.25)))))
   (- -1.5 (* 0.375 (/ (* r w) (/ (/ 1.0 r) w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -2200000000.0) || !(v <= 2.4e-9)) {
		tmp = -1.5 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = -1.5 - (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) :: tmp
    if ((v <= (-2200000000.0d0)) .or. (.not. (v <= 2.4d-9))) then
        tmp = (-1.5d0) - (r * (w * (w * (r * 0.25d0))))
    else
        tmp = (-1.5d0) - (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 tmp;
	if ((v <= -2200000000.0) || !(v <= 2.4e-9)) {
		tmp = -1.5 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = -1.5 - (0.375 * ((r * w) / ((1.0 / r) / w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -2200000000.0) or not (v <= 2.4e-9):
		tmp = -1.5 - (r * (w * (w * (r * 0.25))))
	else:
		tmp = -1.5 - (0.375 * ((r * w) / ((1.0 / r) / w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -2200000000.0) || !(v <= 2.4e-9))
		tmp = Float64(-1.5 - Float64(r * Float64(w * Float64(w * Float64(r * 0.25)))));
	else
		tmp = Float64(-1.5 - 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)
	tmp = 0.0;
	if ((v <= -2200000000.0) || ~((v <= 2.4e-9)))
		tmp = -1.5 - (r * (w * (w * (r * 0.25))));
	else
		tmp = -1.5 - (0.375 * ((r * w) / ((1.0 / r) / w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -2200000000.0], N[Not[LessEqual[v, 2.4e-9]], $MachinePrecision]], N[(-1.5 - N[(r * N[(w * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.2e9 or 2.4e-9 < v

   1. Initial program 80.0%

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

   3. Simplified 95.5%

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

      1. associate-*l/ 88.0%

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

      2. associate-/l* 95.6%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-*l* 95.6%

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

      6. *-un-lft-identity 95.6%

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

      7. times-frac 93.9%

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

      8. /-rgt-identity 93.9%

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

   7. Applied egg-rr 93.9%

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

   8. Taylor expanded in v around inf 95.6%

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

      1. associate-*r* 95.6%

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

   10. Simplified 95.6%

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

   11. Taylor expanded in r around inf 58.1%

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

   if -2.2e9 < v < 2.4e-9

   1. Initial program 90.9%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in v around 0 84.4%

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

      1. *-commutative 84.4%

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

      2. unpow2 84.4%

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

      3. unpow2 84.4%

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

      4. swap-sqr 99.7%

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

      5. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in r around inf 57.6%

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

       1. *-un-lft-identity 57.6%

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

       2. associate-*r* 57.6%

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

       3. metadata-eval 57.6%

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

       4. div-inv 57.6%

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

       5. associate-/r/ 57.6%

          \[\leadsto -1.5 - \color{blue}{\frac{r \cdot w}{\frac{1}{r \cdot w}}} \cdot 0.375 \]

       6. associate-/r* 57.7%

          \[\leadsto -1.5 - \frac{r \cdot w}{\color{blue}{\frac{\frac{1}{r}}{w}}} \cdot 0.375 \]

   11. Applied egg-rr 57.7%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2200000000 \lor \neg \left(v \leq 2.4 \cdot 10^{-9}\right):\\ \;\;\;\;-1.5 - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 - 0.375 \cdot \frac{r \cdot w}{\frac{\frac{1}{r}}{w}}\\ \end{array} \]

Alternative 14: 55.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2100000000 \lor \neg \left(v \leq 4.1 \cdot 10^{-16}\right):\\ \;\;\;\;-1.5 - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 - \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{\frac{1}{r}}{w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -2100000000.0) (not (<= v 4.1e-16)))
   (- -1.5 (* r (* w (* w (* r 0.25)))))
   (- -1.5 (/ (* (* r w) 0.375) (/ (/ 1.0 r) w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -2100000000.0) || !(v <= 4.1e-16)) {
		tmp = -1.5 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = -1.5 - (((r * w) * 0.375) / ((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) :: tmp
    if ((v <= (-2100000000.0d0)) .or. (.not. (v <= 4.1d-16))) then
        tmp = (-1.5d0) - (r * (w * (w * (r * 0.25d0))))
    else
        tmp = (-1.5d0) - (((r * w) * 0.375d0) / ((1.0d0 / r) / w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -2100000000.0) || !(v <= 4.1e-16)) {
		tmp = -1.5 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = -1.5 - (((r * w) * 0.375) / ((1.0 / r) / w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -2100000000.0) or not (v <= 4.1e-16):
		tmp = -1.5 - (r * (w * (w * (r * 0.25))))
	else:
		tmp = -1.5 - (((r * w) * 0.375) / ((1.0 / r) / w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -2100000000.0) || !(v <= 4.1e-16))
		tmp = Float64(-1.5 - Float64(r * Float64(w * Float64(w * Float64(r * 0.25)))));
	else
		tmp = Float64(-1.5 - Float64(Float64(Float64(r * w) * 0.375) / Float64(Float64(1.0 / r) / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -2100000000.0) || ~((v <= 4.1e-16)))
		tmp = -1.5 - (r * (w * (w * (r * 0.25))));
	else
		tmp = -1.5 - (((r * w) * 0.375) / ((1.0 / r) / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -2100000000.0], N[Not[LessEqual[v, 4.1e-16]], $MachinePrecision]], N[(-1.5 - N[(r * N[(w * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.1e9 or 4.10000000000000006e-16 < v

   1. Initial program 80.0%

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

   3. Simplified 95.5%

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

      1. associate-*l/ 88.0%

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

      2. associate-/l* 95.6%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-*l* 95.6%

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

      6. *-un-lft-identity 95.6%

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

      7. times-frac 93.9%

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

      8. /-rgt-identity 93.9%

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

   7. Applied egg-rr 93.9%

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

   8. Taylor expanded in v around inf 95.6%

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

      1. associate-*r* 95.6%

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

   10. Simplified 95.6%

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

   11. Taylor expanded in r around inf 58.1%

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

   if -2.1e9 < v < 4.10000000000000006e-16

   1. Initial program 90.9%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in v around 0 84.4%

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

      1. *-commutative 84.4%

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

      2. unpow2 84.4%

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

      3. unpow2 84.4%

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

      4. swap-sqr 99.7%

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

      5. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in r around inf 57.6%

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

       1. *-commutative 99.7%

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

       2. pow2 99.7%

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

       3. metadata-eval 99.7%

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

       4. pow-div 99.7%

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

       5. pow1 99.7%

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

       6. inv-pow 99.7%

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

       7. associate-*r/ 99.7%

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

       8. associate-/r* 99.8%

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

   11. Applied egg-rr 57.7%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2100000000 \lor \neg \left(v \leq 4.1 \cdot 10^{-16}\right):\\ \;\;\;\;-1.5 - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 - \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{\frac{1}{r}}{w}}\\ \end{array} \]

Alternative 15: 91.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Simplified 96.8%

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

   1. associate-*l/ 92.8%

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

   2. associate-/l* 96.8%

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

   3. associate-*r* 99.8%

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

   4. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-/r/ 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-*l* 96.8%

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

   6. *-un-lft-identity 96.8%

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

   7. times-frac 95.9%

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

   8. /-rgt-identity 95.9%

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

7. Applied egg-rr 95.9%

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

8. Taylor expanded in v around 0 90.9%

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

9. Final simplification 90.9%

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

Alternative 16: 55.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3600000000 \lor \neg \left(v \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;-1.5 - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -3600000000.0) (not (<= v 2.6e-9)))
   (- -1.5 (* r (* w (* w (* r 0.25)))))
   (- -1.5 (* 0.375 (* (* r w) (* r w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -3600000000.0) || !(v <= 2.6e-9)) {
		tmp = -1.5 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = -1.5 - (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) :: tmp
    if ((v <= (-3600000000.0d0)) .or. (.not. (v <= 2.6d-9))) then
        tmp = (-1.5d0) - (r * (w * (w * (r * 0.25d0))))
    else
        tmp = (-1.5d0) - (0.375d0 * ((r * w) * (r * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -3600000000.0) || !(v <= 2.6e-9)) {
		tmp = -1.5 - (r * (w * (w * (r * 0.25))));
	} else {
		tmp = -1.5 - (0.375 * ((r * w) * (r * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -3600000000.0) or not (v <= 2.6e-9):
		tmp = -1.5 - (r * (w * (w * (r * 0.25))))
	else:
		tmp = -1.5 - (0.375 * ((r * w) * (r * w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -3600000000.0) || !(v <= 2.6e-9))
		tmp = Float64(-1.5 - Float64(r * Float64(w * Float64(w * Float64(r * 0.25)))));
	else
		tmp = Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -3600000000.0) || ~((v <= 2.6e-9)))
		tmp = -1.5 - (r * (w * (w * (r * 0.25))));
	else
		tmp = -1.5 - (0.375 * ((r * w) * (r * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -3600000000.0], N[Not[LessEqual[v, 2.6e-9]], $MachinePrecision]], N[(-1.5 - N[(r * N[(w * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -3.6e9 or 2.6000000000000001e-9 < v

   1. Initial program 80.0%

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

   3. Simplified 95.5%

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

      1. associate-*l/ 88.0%

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

      2. associate-/l* 95.6%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r/ 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-*l* 95.6%

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

      6. *-un-lft-identity 95.6%

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

      7. times-frac 93.9%

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

      8. /-rgt-identity 93.9%

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

   7. Applied egg-rr 93.9%

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

   8. Taylor expanded in v around inf 95.6%

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

      1. associate-*r* 95.6%

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

   10. Simplified 95.6%

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

   11. Taylor expanded in r around inf 58.1%

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

   if -3.6e9 < v < 2.6000000000000001e-9

   1. Initial program 90.9%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in v around 0 84.4%

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

      1. *-commutative 84.4%

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

      2. unpow2 84.4%

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

      3. unpow2 84.4%

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

      4. swap-sqr 99.7%

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

      5. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in r around inf 57.6%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3600000000 \lor \neg \left(v \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;-1.5 - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 17: 49.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Simplified 96.8%

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

   1. associate-*l/ 92.8%

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

   2. associate-/l* 96.8%

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

   3. associate-*r* 99.8%

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

   4. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-/r/ 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-*l* 96.8%

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

   6. *-un-lft-identity 96.8%

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

   7. times-frac 95.9%

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

   8. /-rgt-identity 95.9%

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

7. Applied egg-rr 95.9%

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

8. Taylor expanded in v around inf 90.3%

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

   1. associate-*r* 90.3%

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

10. Simplified 90.3%

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

11. Taylor expanded in r around inf 51.8%

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

12. Final simplification 51.8%

    \[\leadsto -1.5 - r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right) \]

Reproduce

herbie shell --seed 2023305 
(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))