Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.1%

Time: 13.6s

Alternatives: 10

Speedup: 1.8×

• 53.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := -1.5 + t\_0\\ \mathbf{if}\;v \leq -8.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(-0.25 \cdot w\right), r \cdot w, t\_1\right)\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(r \cdot w\right) \cdot \left(-0.25 + \frac{0.125}{v}\right), r \cdot w, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ -1.5 t_0)))
   (if (<= v -8.5e+41)
     (fma (* r (* -0.25 w)) (* r w) t_1)
     (if (<= v 1.5)
       (- (- (+ 3.0 t_0) (/ (* 0.375 (* (* r w) (* r w))) (- 1.0 v))) 4.5)
       (fma (* (* r w) (+ -0.25 (/ 0.125 v))) (* r w) t_1)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = -1.5 + t_0;
	double tmp;
	if (v <= -8.5e+41) {
		tmp = fma((r * (-0.25 * w)), (r * w), t_1);
	} else if (v <= 1.5) {
		tmp = ((3.0 + t_0) - ((0.375 * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	} else {
		tmp = fma(((r * w) * (-0.25 + (0.125 / v))), (r * w), t_1);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(-1.5 + t_0)
	tmp = 0.0
	if (v <= -8.5e+41)
		tmp = fma(Float64(r * Float64(-0.25 * w)), Float64(r * w), t_1);
	elseif (v <= 1.5)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(0.375 * Float64(Float64(r * w) * Float64(r * w))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = fma(Float64(Float64(r * w) * Float64(-0.25 + Float64(0.125 / v))), Float64(r * w), t_1);
	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[(-1.5 + t$95$0), $MachinePrecision]}, If[LessEqual[v, -8.5e+41], N[(N[(r * N[(-0.25 * w), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[v, 1.5], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(r * w), $MachinePrecision] * N[(-0.25 + N[(0.125 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -8.49999999999999938e41

   1. Initial program 88.4%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 90.0%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. swap-sqr N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. *-lowering-*.f64 99.8

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 99.8

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

   10. Simplified 99.8%

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

   if -8.49999999999999938e41 < v < 1.5

   1. Initial program 88.9%

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

      1. associate-*l* N/A

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

      2. unswap-sqr N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0

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

   7. Simplified 99.8%

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

   if 1.5 < v

   1. Initial program 77.4%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 85.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. swap-sqr N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. *-lowering-*.f64 99.8

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -8.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(-0.25 \cdot w\right), r \cdot w, -1.5 + \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(r \cdot w\right) \cdot \left(-0.25 + \frac{0.125}{v}\right), r \cdot w, -1.5 + \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (+
       (+ 3.0 (/ 2.0 (* r r)))
       (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* r (* w w)))) (+ v -1.0)))
      5.0)
   (fma (* -0.25 w) (* r (* r w)) -1.5)
   (+ -1.5 (/ (/ 2.0 r) r))))

C

double code(double v, double w, double r) {
	double tmp;
	if (((3.0 + (2.0 / (r * r))) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= 5.0) {
		tmp = fma((-0.25 * w), (r * (r * w)), -1.5);
	} else {
		tmp = -1.5 + ((2.0 / r) / r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= 5.0)
		tmp = fma(Float64(-0.25 * w), Float64(r * Float64(r * w)), -1.5);
	else
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[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[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5.0], N[(N[(-0.25 * w), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $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))) < 5

   1. Initial program 85.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 86.1%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 88.9

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

   7. Applied egg-rr 88.9%

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

   8. Taylor expanded in r around inf

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

   10. Simplified 88.2%

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

   if 5 < (-.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 86.3%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 90.6%

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

   6. Taylor expanded in r around 0

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\frac{-3}{2}} \]
   7. Step-by-step derivation

   8. Simplified 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 99.9

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 93.6%

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

Alternative 3: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* r (* w w)))) (+ v -1.0)))
        5.0)
     (fma (* -0.25 w) (* r (* r w)) -1.5)
     (+ -1.5 t_0))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= 5.0) {
		tmp = fma((-0.25 * w), (r * (r * w)), -1.5);
	} else {
		tmp = -1.5 + t_0;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= 5.0)
		tmp = fma(Float64(-0.25 * w), Float64(r * Float64(r * w)), -1.5);
	else
		tmp = Float64(-1.5 + 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[(N[(3.0 + t$95$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], 5.0], N[(N[(-0.25 * w), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(-1.5 + t$95$0), $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))) < 5

   1. Initial program 85.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 86.1%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 88.9

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

   7. Applied egg-rr 88.9%

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

   8. Taylor expanded in r around inf

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

   10. Simplified 88.2%

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

   if 5 < (-.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 86.3%

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

   3. Taylor expanded in 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 99.8

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

   5. Simplified 99.8%

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

4. Final simplification 93.6%

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

Alternative 4: 89.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -50000:\\ \;\;\;\;r \cdot \left(r \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* r (* w w)))) (+ v -1.0)))
        -50000.0)
     (* r (* r (* -0.25 (* w w))))
     (+ -1.5 t_0))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -50000.0) {
		tmp = r * (r * (-0.25 * (w * w)));
	} else {
		tmp = -1.5 + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) + (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (r * (r * (w * w)))) / (v + (-1.0d0)))) <= (-50000.0d0)) then
        tmp = r * (r * ((-0.25d0) * (w * w)))
    else
        tmp = (-1.5d0) + t_0
    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 (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -50000.0) {
		tmp = r * (r * (-0.25 * (w * w)));
	} else {
		tmp = -1.5 + t_0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -50000.0:
		tmp = r * (r * (-0.25 * (w * w)))
	else:
		tmp = -1.5 + t_0
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= -50000.0)
		tmp = Float64(r * Float64(r * Float64(-0.25 * Float64(w * w))));
	else
		tmp = Float64(-1.5 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -50000.0)
		tmp = r * (r * (-0.25 * (w * w)));
	else
		tmp = -1.5 + t_0;
	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[(3.0 + t$95$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], -50000.0], N[(r * N[(r * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + t$95$0), $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))) < -5e4

   1. Initial program 88.1%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 66.2%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. swap-sqr N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      18. *-lowering-*.f64 71.1

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

   7. Applied egg-rr 71.1%

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

   8. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 93.7

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

   10. Simplified 93.7%

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

   11. Taylor expanded in r around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 88.5

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

   13. Simplified 88.5%

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

   if -5e4 < (-.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 84.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 94.7

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

   5. Simplified 94.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -50000:\\ \;\;\;\;r \cdot \left(r \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 6 \cdot 10^{-50}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{\frac{2}{r}}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, v, 0.375\right), \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 6e-50)
   (+ -1.5 (fma (* (* -0.25 (* r r)) w) w (/ (/ 2.0 r) r)))
   (-
    (+ 3.0 (/ 2.0 (* r r)))
    (fma (fma -0.25 v 0.375) (/ (* r (* w (* r w))) (- 1.0 v)) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 6e-50) {
		tmp = -1.5 + fma(((-0.25 * (r * r)) * w), w, ((2.0 / r) / r));
	} else {
		tmp = (3.0 + (2.0 / (r * r))) - fma(fma(-0.25, v, 0.375), ((r * (w * (r * w))) / (1.0 - v)), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 6e-50)
		tmp = Float64(-1.5 + fma(Float64(Float64(-0.25 * Float64(r * r)) * w), w, Float64(Float64(2.0 / r) / r)));
	else
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - fma(fma(-0.25, v, 0.375), Float64(Float64(r * Float64(w * Float64(r * w))) / Float64(1.0 - v)), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 6e-50], N[(-1.5 + N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.99999999999999981e-50

   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 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 94.1%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 94.1

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

   7. Applied egg-rr 94.1%

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

   if 5.99999999999999981e-50 < r

   1. Initial program 91.2%

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

      1. associate-*l* N/A

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

      2. unswap-sqr N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 92.7

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

   4. Applied egg-rr 92.7%

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

   5. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 92.7

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

   7. Simplified 92.7%

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

      1. associate--l- N/A

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

      2. --lowering--.f64 N/A

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

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

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

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

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

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

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. --lowering--.f64 99.9

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

   9. Applied egg-rr 99.9%

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

Alternative 6: 93.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in v around inf

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

5. Simplified 77.2%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. swap-sqr N/A

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

   6. associate-*r* N/A

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

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

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   18. *-lowering-*.f64 83.3

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

7. Applied egg-rr 83.3%

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

8. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 96.3

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

10. Simplified 96.3%

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

11. Final simplification 96.3%

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

Alternative 7: 86.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 92.5%

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

Alternative 8: 50.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		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 <= 1.15d0) 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 <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		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 <= 1.15)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 84.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 60.0

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

   5. Simplified 60.0%

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

   if 1.1499999999999999 < r

   1. Initial program 91.4%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 67.0%

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

   6. Taylor expanded in r around 0

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\frac{-3}{2}} \]
   7. Step-by-step derivation

   8. Simplified 26.2%

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

   9. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{-3}{2}} \]
   10. Step-by-step derivation

   11. Simplified 26.2%

       \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ -1.5 + \frac{2}{r \cdot r} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 58.2

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

5. Simplified 58.2%

   \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
6. Add Preprocessing

Alternative 10: 13.8% 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 85.9%

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

3. Taylor expanded in v around inf

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

5. Simplified 77.2%

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

6. Taylor expanded in r around 0

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\frac{-3}{2}} \]
7. Step-by-step derivation

8. Simplified 58.2%

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

9. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{-3}{2}} \]
10. Step-by-step derivation

11. Simplified 12.0%

    \[\leadsto \color{blue}{-1.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))