Rosa's TurbineBenchmark

Percentage Accurate: 85.3% → 96.7%

Time: 14.8s

Alternatives: 16

Speedup: 1.6×

• 53.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 + 3\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-321}:\\ \;\;\;\;\left(t\_1 + \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right) - 4.5\\ \mathbf{elif}\;w \cdot w \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\left(t\_1 + \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{r}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.375\right) \cdot w, w, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ t_0 3.0)))
   (if (<= (* w w) 2e-321)
     (- (+ t_1 (/ (* (fma -0.25 v 0.375) (* r (* w (* r w)))) (+ v -1.0))) 4.5)
     (if (<= (* w w) 2e+136)
       (-
        (+ t_1 (* (* (fma v -0.25 0.375) (* r (* w w))) (/ r (+ v -1.0))))
        4.5)
       (fma (* (* (* r r) -0.375) w) w t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 + 3.0;
	double tmp;
	if ((w * w) <= 2e-321) {
		tmp = (t_1 + ((fma(-0.25, v, 0.375) * (r * (w * (r * w)))) / (v + -1.0))) - 4.5;
	} else if ((w * w) <= 2e+136) {
		tmp = (t_1 + ((fma(v, -0.25, 0.375) * (r * (w * w))) * (r / (v + -1.0)))) - 4.5;
	} else {
		tmp = fma((((r * r) * -0.375) * w), w, t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 + 3.0)
	tmp = 0.0
	if (Float64(w * w) <= 2e-321)
		tmp = Float64(Float64(t_1 + Float64(Float64(fma(-0.25, v, 0.375) * Float64(r * Float64(w * Float64(r * w)))) / Float64(v + -1.0))) - 4.5);
	elseif (Float64(w * w) <= 2e+136)
		tmp = Float64(Float64(t_1 + Float64(Float64(fma(v, -0.25, 0.375) * Float64(r * Float64(w * w))) * Float64(r / Float64(v + -1.0)))) - 4.5);
	else
		tmp = fma(Float64(Float64(Float64(r * r) * -0.375) * w), w, t_0);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 3.0), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e-321], N[(N[(t$95$1 + N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[N[(w * w), $MachinePrecision], 2e+136], N[(N[(t$95$1 + N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(N[(r * r), $MachinePrecision] * -0.375), $MachinePrecision] * w), $MachinePrecision] * w + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 w w) < 2.00097e-321

   1. Initial program 87.1%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. accelerator-lowering-fma.f64 87.1

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   5. Simplified 87.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(r \cdot \left(w \cdot w\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. *-lowering-*.f64 96.0

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(\color{blue}{\left(r \cdot w\right)} \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   7. Applied egg-rr 96.0%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)}{1 - v}\right) - 4.5 \]

   if 2.00097e-321 < (*.f64 w w) < 2.00000000000000012e136

   1. Initial program 94.3%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. accelerator-lowering-fma.f64 94.3

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   5. Simplified 94.3%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}}{1 - v}\right) - \frac{9}{2} \]

      2. associate-/l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot \frac{r}{1 - v}}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot \frac{r}{1 - v}}\right) - \frac{9}{2} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\color{blue}{v \cdot \frac{-1}{4}} + \frac{3}{8}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right)} \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot w\right)}\right)\right) \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \color{blue}{\frac{r}{1 - v}}\right) - \frac{9}{2} \]

      11. --lowering--.f64 98.8

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{r}{\color{blue}{1 - v}}\right) - 4.5 \]

   7. Applied egg-rr 98.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{r}{1 - v}}\right) - 4.5 \]

   if 2.00000000000000012e136 < (*.f64 w w)

   1. Initial program 70.1%

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

   3. Taylor expanded in v around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

      15. *-lowering-*.f64 70.5

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

   5. Simplified 70.5%

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

   6. Taylor expanded in w around inf

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot w\right) \cdot \color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot w\right) \cdot \left(\frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}\right) \]

      8. *-lowering-*.f64 70.5

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

   8. Simplified 70.5%

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

   9. Taylor expanded in w around 0

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}} \]

       2. metadata-eval N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {r}^{2}\right) \cdot {w}^{2} + 2 \cdot \frac{1}{{r}^{2}} \]

       3. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}} \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) \cdot w\right) \cdot w + 2 \cdot \frac{1}{{r}^{2}} \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot w\right)\right)} \cdot w + 2 \cdot \frac{1}{{r}^{2}} \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot w\right), w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       16. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \frac{\color{blue}{2}}{{r}^{2}}\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \color{blue}{\frac{2}{{r}^{2}}}\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \frac{2}{\color{blue}{r \cdot r}}\right) \]

       20. *-lowering-*.f64 95.4

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

   11. Simplified 95.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-321}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right) - 4.5\\ \mathbf{elif}\;w \cdot w \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{r}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.375\right) \cdot w, w, \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2000000:\\ \;\;\;\;\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ t_0 3.0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* r (* w w)))) (+ v -1.0)))
        -2000000.0)
     (* (fma v -0.25 0.375) (* (* w (* r w)) (/ r (+ v -1.0))))
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2000000.0) {
		tmp = fma(v, -0.25, 0.375) * ((w * (r * w)) * (r / (v + -1.0)));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= -2000000.0)
		tmp = Float64(fma(v, -0.25, 0.375) * Float64(Float64(w * Float64(r * w)) * Float64(r / Float64(v + -1.0))));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000.0], N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -2e6

   1. Initial program 81.6%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. accelerator-lowering-fma.f64 81.6

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   5. Simplified 81.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{1 - v}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left({r}^{2} \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {r}^{2}\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {r}^{2}\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v}} \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({r}^{2}\right)\right)} \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({r}^{2}\right)\right)} \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{r \cdot r}\right)\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{r \cdot r}\right)\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{\frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \frac{\color{blue}{\frac{-1}{4} \cdot v + \frac{3}{8}}}{1 - v}\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right)}}{1 - v}\right) \]

      15. --lowering--.f64 81.7

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

   8. Simplified 81.7%

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{\frac{-1}{4} \cdot v + \frac{3}{8}}{1 - v}} \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\right)} \cdot \frac{\frac{-1}{4} \cdot v + \frac{3}{8}}{1 - v} \]

      3. swap-sqr N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}\right)\right) \cdot \frac{\frac{-1}{4} \cdot v + \frac{3}{8}}{1 - v} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(r \cdot w\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \cdot \frac{\frac{-1}{4} \cdot v + \frac{3}{8}}{1 - v} \]

      5. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right) \cdot r}\right)\right) \cdot \frac{\frac{-1}{4} \cdot v + \frac{3}{8}}{1 - v} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)\right) \cdot \left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)}{1 - v}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right) \cdot \left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)\right)}}{1 - v} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}\right)}{1 - v} \]

      9. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}{1 - v}\right)} \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}{\mathsf{neg}\left(\left(1 - v\right)\right)}} \]

      11. neg-mul-1 N/A

          \[\leadsto \frac{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}{\color{blue}{-1 \cdot \left(1 - v\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)\right) \cdot r}}{-1 \cdot \left(1 - v\right)} \]

      13. times-frac N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)}{-1} \cdot \frac{r}{1 - v}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)}{-1} \cdot \frac{r}{1 - v}} \]

   10. Applied egg-rr 83.6%

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

       1. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{-1}\right)} \cdot \frac{r}{1 - v} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\frac{r \cdot \left(w \cdot w\right)}{-1} \cdot \frac{r}{1 - v}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\frac{r \cdot \left(w \cdot w\right)}{-1} \cdot \frac{r}{1 - v}\right)} \]

       4. *-commutative N/A

          \[\leadsto \left(\color{blue}{v \cdot \frac{-1}{4}} + \frac{3}{8}\right) \cdot \left(\frac{r \cdot \left(w \cdot w\right)}{-1} \cdot \frac{r}{1 - v}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right)} \cdot \left(\frac{r \cdot \left(w \cdot w\right)}{-1} \cdot \frac{r}{1 - v}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \color{blue}{\left(\frac{r \cdot \left(w \cdot w\right)}{-1} \cdot \frac{r}{1 - v}\right)} \]

       7. div-inv N/A

          \[\leadsto \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \frac{1}{-1}\right)} \cdot \frac{r}{1 - v}\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{-1}\right) \cdot \frac{r}{1 - v}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot -1\right)} \cdot \frac{r}{1 - v}\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot -1\right) \cdot \frac{r}{1 - v}\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot -1\right) \cdot \frac{r}{1 - v}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot -1\right) \cdot \frac{r}{1 - v}\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot -1\right) \cdot \frac{r}{1 - v}\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot -1\right) \cdot \frac{r}{1 - v}\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\left(\left(w \cdot \left(w \cdot r\right)\right) \cdot -1\right) \cdot \color{blue}{\frac{r}{1 - v}}\right) \]

       16. --lowering--.f64 90.5

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

   12. Applied egg-rr 90.5%

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

   if -2e6 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 84.1%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 96.3

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Simplified 96.3%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2000000:\\ \;\;\;\;\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2000000:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.375\right) \cdot w, w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ t_0 3.0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* r (* w w)))) (+ v -1.0)))
        -2000000.0)
     (fma (* (* (* r r) -0.375) w) w t_0)
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2000000.0) {
		tmp = fma((((r * r) * -0.375) * w), w, t_0);
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= -2000000.0)
		tmp = fma(Float64(Float64(Float64(r * r) * -0.375) * w), w, t_0);
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000.0], N[(N[(N[(N[(r * r), $MachinePrecision] * -0.375), $MachinePrecision] * w), $MachinePrecision] * w + t$95$0), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -2e6

   1. Initial program 81.6%

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

   3. Taylor expanded in v around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

      15. *-lowering-*.f64 75.3

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

   5. Simplified 75.3%

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

   6. Taylor expanded in w around inf

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot w\right) \cdot \color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot w\right) \cdot \left(\frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}\right) \]

      8. *-lowering-*.f64 74.9

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

   8. Simplified 74.9%

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

   9. Taylor expanded in w around 0

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}} \]

       2. metadata-eval N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {r}^{2}\right) \cdot {w}^{2} + 2 \cdot \frac{1}{{r}^{2}} \]

       3. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}} \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) \cdot w\right) \cdot w + 2 \cdot \frac{1}{{r}^{2}} \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot w\right)\right)} \cdot w + 2 \cdot \frac{1}{{r}^{2}} \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot w\right), w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       16. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \frac{\color{blue}{2}}{{r}^{2}}\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \color{blue}{\frac{2}{{r}^{2}}}\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \frac{2}{\color{blue}{r \cdot r}}\right) \]

       20. *-lowering-*.f64 82.7

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

   11. Simplified 82.7%

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

   if -2e6 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 84.1%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 96.3

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Simplified 96.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2000000:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.375\right) \cdot w, w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2000000:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ t_0 3.0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* r (* w w)))) (+ v -1.0)))
        -2000000.0)
     (* (* r r) (* (* w w) -0.25))
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2000000.0) {
		tmp = (r * r) * ((w * w) * -0.25);
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) + (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (r * (r * (w * w)))) / (v + (-1.0d0)))) <= (-2000000.0d0)) then
        tmp = (r * r) * ((w * w) * (-0.25d0))
    else
        tmp = t_0 + (-1.5d0)
    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 (((t_0 + 3.0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2000000.0) {
		tmp = (r * r) * ((w * w) * -0.25);
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((t_0 + 3.0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2000000.0:
		tmp = (r * r) * ((w * w) * -0.25)
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= -2000000.0)
		tmp = Float64(Float64(r * r) * Float64(Float64(w * w) * -0.25));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2000000.0)
		tmp = (r * r) * ((w * w) * -0.25);
	else
		tmp = t_0 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000.0], N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -2e6

   1. Initial program 81.6%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. accelerator-lowering-fma.f64 81.6

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   5. Simplified 81.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{1 - v}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left({r}^{2} \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {r}^{2}\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {r}^{2}\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v}} \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({r}^{2}\right)\right)} \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({r}^{2}\right)\right)} \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{r \cdot r}\right)\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{r \cdot r}\right)\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{\frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \frac{\color{blue}{\frac{-1}{4} \cdot v + \frac{3}{8}}}{1 - v}\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right)}}{1 - v}\right) \]

      15. --lowering--.f64 81.7

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

   8. Simplified 81.7%

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

   9. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-1}{4}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left({w}^{2} \cdot \frac{-1}{4}\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-1}{4} \cdot {w}^{2}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)} \]

       9. unpow2 N/A

          \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \frac{-1}{4}\right) \]

       10. *-lowering-*.f64 77.3

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

   11. Simplified 77.3%

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

   if -2e6 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 84.1%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 96.3

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Simplified 96.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2000000:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2000000:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ t_0 3.0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* r (* w w)))) (+ v -1.0)))
        -2000000.0)
     (* (* r r) (* -0.375 (* w w)))
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2000000.0) {
		tmp = (r * r) * (-0.375 * (w * w));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) + (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (r * (r * (w * w)))) / (v + (-1.0d0)))) <= (-2000000.0d0)) then
        tmp = (r * r) * ((-0.375d0) * (w * w))
    else
        tmp = t_0 + (-1.5d0)
    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 (((t_0 + 3.0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2000000.0) {
		tmp = (r * r) * (-0.375 * (w * w));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((t_0 + 3.0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2000000.0:
		tmp = (r * r) * (-0.375 * (w * w))
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= -2000000.0)
		tmp = Float64(Float64(r * r) * Float64(-0.375 * Float64(w * w)));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2000000.0)
		tmp = (r * r) * (-0.375 * (w * w));
	else
		tmp = t_0 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000.0], N[(N[(r * r), $MachinePrecision] * N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -2e6

   1. Initial program 81.6%

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

   3. Taylor expanded in v around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

      15. *-lowering-*.f64 75.3

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

   5. Simplified 75.3%

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

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot {w}^{2} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)} \]

      4. metadata-eval N/A

         \[\leadsto {r}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {w}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{r}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {w}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \color{blue}{\left(r \cdot r\right)} \cdot \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {w}^{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(r \cdot r\right)} \cdot \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {w}^{2}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2}\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(r \cdot r\right) \cdot \left(\frac{-3}{8} \cdot \color{blue}{\left(w \cdot w\right)}\right) \]

      11. *-lowering-*.f64 74.9

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

   8. Simplified 74.9%

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

   if -2e6 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 84.1%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 96.3

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Simplified 96.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2000000:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 4.3 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.375\right) \cdot w, w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) + \left(0.125 \cdot \left(w \cdot \mathsf{fma}\left(v, -2, 3\right)\right)\right) \cdot \left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \frac{1}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 4.3e-113)
     (fma (* (* (* r r) -0.375) w) w t_0)
     (-
      (+
       (+ t_0 3.0)
       (*
        (* 0.125 (* w (fma v -2.0 3.0)))
        (* (* r (* r w)) (/ 1.0 (+ v -1.0)))))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 4.3e-113) {
		tmp = fma((((r * r) * -0.375) * w), w, t_0);
	} else {
		tmp = ((t_0 + 3.0) + ((0.125 * (w * fma(v, -2.0, 3.0))) * ((r * (r * w)) * (1.0 / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 4.3e-113)
		tmp = fma(Float64(Float64(Float64(r * r) * -0.375) * w), w, t_0);
	else
		tmp = Float64(Float64(Float64(t_0 + 3.0) + Float64(Float64(0.125 * Float64(w * fma(v, -2.0, 3.0))) * Float64(Float64(r * Float64(r * w)) * Float64(1.0 / Float64(v + -1.0))))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 4.3e-113], N[(N[(N[(N[(r * r), $MachinePrecision] * -0.375), $MachinePrecision] * w), $MachinePrecision] * w + t$95$0), $MachinePrecision], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(0.125 * N[(w * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4.3e-113

   1. Initial program 78.3%

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

   3. Taylor expanded in v around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

      15. *-lowering-*.f64 73.3

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

   5. Simplified 73.3%

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

   6. Taylor expanded in w around inf

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot w\right) \cdot \color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot w\right) \cdot \left(\frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}\right) \]

      8. *-lowering-*.f64 68.5

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

   8. Simplified 68.5%

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

   9. Taylor expanded in w around 0

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}} \]

       2. metadata-eval N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {r}^{2}\right) \cdot {w}^{2} + 2 \cdot \frac{1}{{r}^{2}} \]

       3. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}} \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) \cdot w\right) \cdot w + 2 \cdot \frac{1}{{r}^{2}} \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot w\right)\right)} \cdot w + 2 \cdot \frac{1}{{r}^{2}} \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot w\right), w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       16. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \frac{\color{blue}{2}}{{r}^{2}}\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \color{blue}{\frac{2}{{r}^{2}}}\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \frac{2}{\color{blue}{r \cdot r}}\right) \]

       20. *-lowering-*.f64 82.3

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

   11. Simplified 82.3%

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

   if 4.3e-113 < r

   1. Initial program 92.2%

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

      1. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(w \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot r\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(w \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. *-lowering-*.f64 95.5

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

   4. Applied egg-rr 95.5%

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

      1. div-inv N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot r\right)\right)\right) \cdot \frac{1}{1 - v}}\right) - \frac{9}{2} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot r\right)\right)} \cdot \frac{1}{1 - v}\right) - \frac{9}{2} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right) \cdot \left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)}\right) - \frac{9}{2} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right) \cdot \left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)}\right) - \frac{9}{2} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\frac{1}{8} \cdot \color{blue}{\left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)}\right) \cdot w\right) \cdot \left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\frac{1}{8} \cdot \left(3 + \color{blue}{-2} \cdot v\right)\right) \cdot w\right) \cdot \left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\frac{1}{8} \cdot \left(3 + \color{blue}{v \cdot -2}\right)\right) \cdot w\right) \cdot \left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      8. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\frac{1}{8} \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot w\right) \cdot \left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{1}{8} \cdot \left(\left(v \cdot -2 + 3\right) \cdot w\right)\right)} \cdot \left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{1}{8} \cdot \left(\left(v \cdot -2 + 3\right) \cdot w\right)\right)} \cdot \left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \color{blue}{\left(\left(v \cdot -2 + 3\right) \cdot w\right)}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\color{blue}{\mathsf{fma}\left(v, -2, 3\right)} \cdot w\right)\right) \cdot \left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right)\right) \cdot \color{blue}{\left(\left(\left(r \cdot w\right) \cdot r\right) \cdot \frac{1}{1 - v}\right)}\right) - \frac{9}{2} \]

      14. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right)\right) \cdot \left(\color{blue}{\left(r \cdot \left(r \cdot w\right)\right)} \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right)\right) \cdot \left(\color{blue}{\left(r \cdot \left(r \cdot w\right)\right)} \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right)\right) \cdot \left(\left(r \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot \frac{1}{1 - v}\right)\right) - \frac{9}{2} \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right)\right) \cdot \left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\frac{1}{1 - v}}\right)\right) - \frac{9}{2} \]

      18. --lowering--.f64 97.6

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

   6. Applied egg-rr 97.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4.3 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.375\right) \cdot w, w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \left(0.125 \cdot \left(w \cdot \mathsf{fma}\left(v, -2, 3\right)\right)\right) \cdot \left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \frac{1}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.375\right) \cdot w, w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) + \left(0.125 \cdot \left(w \cdot \mathsf{fma}\left(v, -2, 3\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot w\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 5e-113)
     (fma (* (* (* r r) -0.375) w) w t_0)
     (-
      (+
       (+ t_0 3.0)
       (* (* 0.125 (* w (fma v -2.0 3.0))) (/ (* r (* r w)) (+ v -1.0))))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 5e-113) {
		tmp = fma((((r * r) * -0.375) * w), w, t_0);
	} else {
		tmp = ((t_0 + 3.0) + ((0.125 * (w * fma(v, -2.0, 3.0))) * ((r * (r * w)) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 5e-113)
		tmp = fma(Float64(Float64(Float64(r * r) * -0.375) * w), w, t_0);
	else
		tmp = Float64(Float64(Float64(t_0 + 3.0) + Float64(Float64(0.125 * Float64(w * fma(v, -2.0, 3.0))) * Float64(Float64(r * Float64(r * w)) / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 5e-113], N[(N[(N[(N[(r * r), $MachinePrecision] * -0.375), $MachinePrecision] * w), $MachinePrecision] * w + t$95$0), $MachinePrecision], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(0.125 * N[(w * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4.9999999999999997e-113

   1. Initial program 78.3%

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

   3. Taylor expanded in v around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

      15. *-lowering-*.f64 73.3

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

   5. Simplified 73.3%

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

   6. Taylor expanded in w around inf

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot w\right) \cdot \color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot w\right) \cdot \left(\frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}\right) \]

      8. *-lowering-*.f64 68.5

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

   8. Simplified 68.5%

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

   9. Taylor expanded in w around 0

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}} \]

       2. metadata-eval N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {r}^{2}\right) \cdot {w}^{2} + 2 \cdot \frac{1}{{r}^{2}} \]

       3. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}} \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) \cdot w\right) \cdot w + 2 \cdot \frac{1}{{r}^{2}} \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot w\right)\right)} \cdot w + 2 \cdot \frac{1}{{r}^{2}} \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot w\right), w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       16. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \frac{\color{blue}{2}}{{r}^{2}}\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \color{blue}{\frac{2}{{r}^{2}}}\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \frac{2}{\color{blue}{r \cdot r}}\right) \]

       20. *-lowering-*.f64 82.3

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

   11. Simplified 82.3%

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

   if 4.9999999999999997e-113 < r

   1. Initial program 92.2%

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

      1. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(w \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot r\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(w \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. *-lowering-*.f64 95.5

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

   4. Applied egg-rr 95.5%

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

      1. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. associate-/l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot r}{1 - v}}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot r}{1 - v}}\right) - \frac{9}{2} \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\frac{1}{8} \cdot \color{blue}{\left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)}\right) \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\frac{1}{8} \cdot \left(3 + \color{blue}{-2} \cdot v\right)\right) \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\frac{1}{8} \cdot \left(3 + \color{blue}{v \cdot -2}\right)\right) \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      7. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\frac{1}{8} \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{1}{8} \cdot \left(\left(v \cdot -2 + 3\right) \cdot w\right)\right)} \cdot \frac{\left(r \cdot w\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{1}{8} \cdot \left(\left(v \cdot -2 + 3\right) \cdot w\right)\right)} \cdot \frac{\left(r \cdot w\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \color{blue}{\left(\left(v \cdot -2 + 3\right) \cdot w\right)}\right) \cdot \frac{\left(r \cdot w\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\color{blue}{\mathsf{fma}\left(v, -2, 3\right)} \cdot w\right)\right) \cdot \frac{\left(r \cdot w\right) \cdot r}{1 - v}\right) - \frac{9}{2} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right)\right) \cdot \color{blue}{\frac{\left(r \cdot w\right) \cdot r}{1 - v}}\right) - \frac{9}{2} \]

      13. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right)\right) \cdot \frac{\color{blue}{r \cdot \left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right)\right) \cdot \frac{\color{blue}{r \cdot \left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{1}{8} \cdot \left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right)\right) \cdot \frac{r \cdot \color{blue}{\left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      16. --lowering--.f64 97.6

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

   6. Applied egg-rr 97.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.375\right) \cdot w, w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \left(0.125 \cdot \left(w \cdot \mathsf{fma}\left(v, -2, 3\right)\right)\right) \cdot \frac{r \cdot \left(r \cdot w\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) + \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{r}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.375\right) \cdot w, w, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e+136)
     (-
      (+
       (+ t_0 3.0)
       (* (* (fma v -0.25 0.375) (* r (* w w))) (/ r (+ v -1.0))))
      4.5)
     (fma (* (* (* r r) -0.375) w) w t_0))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+136) {
		tmp = ((t_0 + 3.0) + ((fma(v, -0.25, 0.375) * (r * (w * w))) * (r / (v + -1.0)))) - 4.5;
	} else {
		tmp = fma((((r * r) * -0.375) * w), w, t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e+136)
		tmp = Float64(Float64(Float64(t_0 + 3.0) + Float64(Float64(fma(v, -0.25, 0.375) * Float64(r * Float64(w * w))) * Float64(r / Float64(v + -1.0)))) - 4.5);
	else
		tmp = fma(Float64(Float64(Float64(r * r) * -0.375) * w), w, t_0);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e+136], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(N[(r * r), $MachinePrecision] * -0.375), $MachinePrecision] * w), $MachinePrecision] * w + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 2.00000000000000012e136

   1. Initial program 91.9%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. accelerator-lowering-fma.f64 91.9

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   5. Simplified 91.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}}{1 - v}\right) - \frac{9}{2} \]

      2. associate-/l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot \frac{r}{1 - v}}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot \frac{r}{1 - v}}\right) - \frac{9}{2} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\color{blue}{v \cdot \frac{-1}{4}} + \frac{3}{8}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right)} \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot w\right)}\right)\right) \cdot \frac{r}{1 - v}\right) - \frac{9}{2} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \color{blue}{\frac{r}{1 - v}}\right) - \frac{9}{2} \]

      11. --lowering--.f64 95.1

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{r}{\color{blue}{1 - v}}\right) - 4.5 \]

   7. Applied egg-rr 95.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{r}{1 - v}}\right) - 4.5 \]

   if 2.00000000000000012e136 < (*.f64 w w)

   1. Initial program 70.1%

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

   3. Taylor expanded in v around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

      15. *-lowering-*.f64 70.5

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

   5. Simplified 70.5%

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

   6. Taylor expanded in w around inf

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot w\right) \cdot \color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot w\right) \cdot \left(\frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}\right) \]

      8. *-lowering-*.f64 70.5

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

   8. Simplified 70.5%

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

   9. Taylor expanded in w around 0

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}} \]

       2. metadata-eval N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {r}^{2}\right) \cdot {w}^{2} + 2 \cdot \frac{1}{{r}^{2}} \]

       3. unpow2 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}} \]

       5. metadata-eval N/A

          \[\leadsto \left(\left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) \cdot w\right) \cdot w + 2 \cdot \frac{1}{{r}^{2}} \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot w\right)\right)} \cdot w + 2 \cdot \frac{1}{{r}^{2}} \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot w\right), w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{3}{8}\right)\right)} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left({r}^{2} \cdot \frac{-3}{8}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

       16. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \frac{\color{blue}{2}}{{r}^{2}}\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \color{blue}{\frac{2}{{r}^{2}}}\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot w, w, \frac{2}{\color{blue}{r \cdot r}}\right) \]

       20. *-lowering-*.f64 95.4

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

   11. Simplified 95.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{r}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.375\right) \cdot w, w, \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 22000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot 0.375\right), w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 22000.0)
   (- (/ 2.0 (* r r)) (fma (* w (* (* r r) 0.375)) w 1.5))
   (- (+ 3.0 (/ (* (fma -0.25 v 0.375) (* r (* w (* r w)))) (+ v -1.0))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 22000.0) {
		tmp = (2.0 / (r * r)) - fma((w * ((r * r) * 0.375)), w, 1.5);
	} else {
		tmp = (3.0 + ((fma(-0.25, v, 0.375) * (r * (w * (r * w)))) / (v + -1.0))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 22000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(w * Float64(Float64(r * r) * 0.375)), w, 1.5));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(fma(-0.25, v, 0.375) * Float64(r * Float64(w * Float64(r * w)))) / Float64(v + -1.0))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 22000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(w * N[(N[(r * r), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 22000

   1. Initial program 79.3%

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

   3. Taylor expanded in v around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

      15. *-lowering-*.f64 75.0

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

   5. Simplified 75.0%

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

      1. associate-*l* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{w \cdot \left(w \cdot \left(\frac{3}{8} \cdot \left(r \cdot r\right)\right)\right)} + \frac{3}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(w \cdot \left(\frac{3}{8} \cdot \left(r \cdot r\right)\right)\right) \cdot w} + \frac{3}{2}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left(w \cdot \left(\frac{3}{8} \cdot \left(r \cdot r\right)\right), w, \frac{3}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot \left(\frac{3}{8} \cdot \left(r \cdot r\right)\right)}, w, \frac{3}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{3}{8}\right)}, w, \frac{3}{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{3}{8}\right)}, w, \frac{3}{2}\right) \]

      7. *-lowering-*.f64 88.4

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

   7. Applied egg-rr 88.4%

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

   if 22000 < r

   1. Initial program 95.1%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. accelerator-lowering-fma.f64 95.1

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   5. Simplified 95.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(r \cdot \left(w \cdot w\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. *-lowering-*.f64 96.7

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(\color{blue}{\left(r \cdot w\right)} \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   7. Applied egg-rr 96.7%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)}{1 - v}\right) - 4.5 \]

   8. Taylor expanded in r around inf

      \[\leadsto \left(\color{blue}{3} - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   9. Step-by-step derivation

   10. Simplified 96.7%

       \[\leadsto \left(\color{blue}{3} - \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 22000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot 0.375\right), w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-84}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(r \cdot 0.375\right), r, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e-84)
     (- t_0 (fma (* (* w w) (* r 0.375)) r 1.5))
     (+ -1.5 (fma (* w (* (* r r) -0.25)) w t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e-84) {
		tmp = t_0 - fma(((w * w) * (r * 0.375)), r, 1.5);
	} else {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e-84)
		tmp = Float64(t_0 - fma(Float64(Float64(w * w) * Float64(r * 0.375)), r, 1.5));
	else
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, t_0));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e-84], N[(t$95$0 - N[(N[(N[(w * w), $MachinePrecision] * N[(r * 0.375), $MachinePrecision]), $MachinePrecision] * r + 1.5), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 2.0000000000000001e-84

   1. Initial program 89.0%

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

   3. Taylor expanded in v around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

      15. *-lowering-*.f64 77.2

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

   5. Simplified 77.2%

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

      1. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(\frac{3}{8} \cdot r\right) \cdot r\right)} + \frac{3}{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} \cdot r\right)\right) \cdot r} + \frac{3}{2}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} \cdot r\right), r, \frac{3}{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{\left(w \cdot w\right) \cdot \left(\frac{3}{8} \cdot r\right)}, r, \frac{3}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{3}{8} \cdot r\right), r, \frac{3}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\left(w \cdot w\right) \cdot \color{blue}{\left(r \cdot \frac{3}{8}\right)}, r, \frac{3}{2}\right) \]

      7. *-lowering-*.f64 87.2

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

   7. Applied egg-rr 87.2%

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

   if 2.0000000000000001e-84 < (*.f64 w w)

   1. Initial program 78.9%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 95.2%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(r \cdot 0.375\right), r, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-84}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(r \cdot 0.375, r \cdot \left(w \cdot w\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e-84)
     (- t_0 (fma (* r 0.375) (* r (* w w)) 1.5))
     (+ -1.5 (fma (* w (* (* r r) -0.25)) w t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e-84) {
		tmp = t_0 - fma((r * 0.375), (r * (w * w)), 1.5);
	} else {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e-84)
		tmp = Float64(t_0 - fma(Float64(r * 0.375), Float64(r * Float64(w * w)), 1.5));
	else
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, t_0));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e-84], N[(t$95$0 - N[(N[(r * 0.375), $MachinePrecision] * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 2.0000000000000001e-84

   1. Initial program 89.0%

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

   3. Taylor expanded in v around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

      15. *-lowering-*.f64 77.2

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

   5. Simplified 77.2%

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

      1. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)} + \frac{3}{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\left(\frac{3}{8} \cdot r\right) \cdot r\right)} \cdot \left(w \cdot w\right) + \frac{3}{2}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)} + \frac{3}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\left(\frac{3}{8} \cdot r\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} + \frac{3}{2}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left(\frac{3}{8} \cdot r, \left(w \cdot w\right) \cdot r, \frac{3}{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{r \cdot \frac{3}{8}}, \left(w \cdot w\right) \cdot r, \frac{3}{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{r \cdot \frac{3}{8}}, \left(w \cdot w\right) \cdot r, \frac{3}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(r \cdot \frac{3}{8}, \color{blue}{r \cdot \left(w \cdot w\right)}, \frac{3}{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(r \cdot \frac{3}{8}, \color{blue}{r \cdot \left(w \cdot w\right)}, \frac{3}{2}\right) \]

      10. *-lowering-*.f64 87.2

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

   7. Applied egg-rr 87.2%

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

   if 2.0000000000000001e-84 < (*.f64 w w)

   1. Initial program 78.9%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 95.2%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r \cdot 0.375, r \cdot \left(w \cdot w\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{-276}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 5e-276)
     (+ t_0 -1.5)
     (+ -1.5 (fma (* w (* (* r r) -0.25)) w t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 5e-276) {
		tmp = t_0 + -1.5;
	} else {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 5e-276)
		tmp = Float64(t_0 + -1.5);
	else
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, t_0));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 5e-276], N[(t$95$0 + -1.5), $MachinePrecision], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 4.99999999999999967e-276

   1. Initial program 89.2%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 85.7

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Simplified 85.7%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]

   if 4.99999999999999967e-276 < (*.f64 w w)

   1. Initial program 81.0%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 92.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{-276}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.9 \cdot 10^{+153}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{r}{v + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.9e+153)
   (+ -1.5 (fma (* w (* (* r r) -0.25)) w (/ 2.0 (* r r))))
   (* (* (fma v -0.25 0.375) (* w (* r w))) (/ r (+ v -1.0)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.9e+153) {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, (2.0 / (r * r)));
	} else {
		tmp = (fma(v, -0.25, 0.375) * (w * (r * w))) * (r / (v + -1.0));
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.9e+153)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, Float64(2.0 / Float64(r * r))));
	else
		tmp = Float64(Float64(fma(v, -0.25, 0.375) * Float64(w * Float64(r * w))) * Float64(r / Float64(v + -1.0)));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.9e+153], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.89999999999999983e153

   1. Initial program 81.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 88.8%

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

   if 1.89999999999999983e153 < r

   1. Initial program 93.6%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. accelerator-lowering-fma.f64 93.6

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   5. Simplified 93.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(-0.25, v, 0.375\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{1 - v}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left({r}^{2} \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {r}^{2}\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot {r}^{2}\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v}} \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({r}^{2}\right)\right)} \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({r}^{2}\right)\right)} \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{r \cdot r}\right)\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{r \cdot r}\right)\right) \cdot \frac{{w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{1 - v} \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \color{blue}{\left({w}^{2} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{\frac{\frac{3}{8} + \frac{-1}{4} \cdot v}{1 - v}}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \frac{\color{blue}{\frac{-1}{4} \cdot v + \frac{3}{8}}}{1 - v}\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right)}}{1 - v}\right) \]

      15. --lowering--.f64 80.3

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

   8. Simplified 80.3%

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{\frac{-1}{4} \cdot v + \frac{3}{8}}{1 - v}} \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\right)} \cdot \frac{\frac{-1}{4} \cdot v + \frac{3}{8}}{1 - v} \]

      3. swap-sqr N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}\right)\right) \cdot \frac{\frac{-1}{4} \cdot v + \frac{3}{8}}{1 - v} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(r \cdot w\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \cdot \frac{\frac{-1}{4} \cdot v + \frac{3}{8}}{1 - v} \]

      5. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right) \cdot r}\right)\right) \cdot \frac{\frac{-1}{4} \cdot v + \frac{3}{8}}{1 - v} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)\right) \cdot \left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)}{1 - v}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right) \cdot \left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)\right)}}{1 - v} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}\right)}{1 - v} \]

      9. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}{1 - v}\right)} \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}{\mathsf{neg}\left(\left(1 - v\right)\right)}} \]

      11. neg-mul-1 N/A

          \[\leadsto \frac{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}{\color{blue}{-1 \cdot \left(1 - v\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)\right) \cdot r}}{-1 \cdot \left(1 - v\right)} \]

      13. times-frac N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)}{-1} \cdot \frac{r}{1 - v}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)}{-1} \cdot \frac{r}{1 - v}} \]

   10. Applied egg-rr 87.0%

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

       1. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{\mathsf{neg}\left(-1\right)}} \cdot \frac{r}{1 - v} \]

       2. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{\color{blue}{1}} \cdot \frac{r}{1 - v} \]

       3. /-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)} \cdot \frac{r}{1 - v} \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\mathsf{neg}\left(r \cdot \left(w \cdot w\right)\right)\right)\right)} \cdot \frac{r}{1 - v} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right) \cdot \left(\mathsf{neg}\left(r \cdot \left(w \cdot w\right)\right)\right)\right)} \cdot \frac{r}{1 - v} \]

       6. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{v \cdot \frac{-1}{4}} + \frac{3}{8}\right) \cdot \left(\mathsf{neg}\left(r \cdot \left(w \cdot w\right)\right)\right)\right) \cdot \frac{r}{1 - v} \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right)} \cdot \left(\mathsf{neg}\left(r \cdot \left(w \cdot w\right)\right)\right)\right) \cdot \frac{r}{1 - v} \]

       8. neg-lowering-neg.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(r \cdot \left(w \cdot w\right)\right)\right)}\right) \cdot \frac{r}{1 - v} \]

       9. associate-*r* N/A

          \[\leadsto \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(r \cdot w\right) \cdot w}\right)\right)\right) \cdot \frac{r}{1 - v} \]

       10. *-commutative N/A

           \[\leadsto \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{w \cdot \left(r \cdot w\right)}\right)\right)\right) \cdot \frac{r}{1 - v} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{w \cdot \left(r \cdot w\right)}\right)\right)\right) \cdot \frac{r}{1 - v} \]

       12. *-commutative N/A

           \[\leadsto \left(\mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right) \cdot \left(\mathsf{neg}\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right)\right)\right) \cdot \frac{r}{1 - v} \]

       13. *-lowering-*.f64 90.7

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

   12. Applied egg-rr 90.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.9 \cdot 10^{+153}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{r}{v + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.2:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.2) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.2) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.2d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.2) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.2:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.2)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.2)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.2], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 0.20000000000000001

   1. Initial program 79.1%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

      3. *-lowering-*.f64 56.4

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

   5. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]

   if 0.20000000000000001 < r

   1. Initial program 95.3%

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

   3. Taylor expanded in v around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

      15. *-lowering-*.f64 83.8

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

   5. Simplified 83.8%

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

   6. Taylor expanded in w around 0

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\frac{3}{2}} \]
   7. Step-by-step derivation

   8. Simplified 28.8%

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{1.5} \]

   9. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{-3}{2}} \]
   10. Step-by-step derivation

   11. Simplified 27.8%

       \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -1.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

   9. *-lowering-*.f64 55.6

      \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

5. Simplified 55.6%

   \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]

6. Final simplification 55.6%

   \[\leadsto \frac{2}{r \cdot r} + -1.5 \]
7. Add Preprocessing

Alternative 16: 13.6% accurate, 73.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

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

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in v around 0

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

   1. --lowering--.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   2. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)} \]

   8. associate-*r* N/A

      \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}} + \frac{3}{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)} + \frac{3}{2}\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(\color{blue}{w \cdot w}, \frac{3}{8} \cdot {r}^{2}, \frac{3}{2}\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \color{blue}{\frac{3}{8} \cdot {r}^{2}}, \frac{3}{2}\right) \]

   14. unpow2 N/A

       \[\leadsto \frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot w, \frac{3}{8} \cdot \color{blue}{\left(r \cdot r\right)}, \frac{3}{2}\right) \]

   15. *-lowering-*.f64 76.9

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

5. Simplified 76.9%

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

6. Taylor expanded in w around 0

   \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\frac{3}{2}} \]
7. Step-by-step derivation

8. Simplified 55.6%

   \[\leadsto \frac{2}{r \cdot r} - \color{blue}{1.5} \]

9. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{-3}{2}} \]
10. Step-by-step derivation

11. Simplified 12.8%

    \[\leadsto \color{blue}{-1.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))