Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 96.4%

Time: 11.3s

Alternatives: 7

Speedup: 1.8×

• 54.0% 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: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 3 + t\_0\\ t_2 := -1.5 + \mathsf{fma}\left(w, -0.25 \cdot \left(r \cdot \left(r \cdot w\right)\right), t\_0\right)\\ t_3 := r \cdot \left(r \cdot \left(w \cdot w\right)\right)\\ t_4 := t\_1 + \frac{t\_3 \cdot \left(0.125 \cdot \left(2 \cdot v - 3\right)\right)}{1 - v}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\left(t\_1 + \frac{t\_3 \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (+ 3.0 t_0))
        (t_2 (+ -1.5 (fma w (* -0.25 (* r (* r w))) t_0)))
        (t_3 (* r (* r (* w w))))
        (t_4 (+ t_1 (/ (* t_3 (* 0.125 (- (* 2.0 v) 3.0))) (- 1.0 v)))))
   (if (<= t_4 (- INFINITY))
     t_2
     (if (<= t_4 -2e+16)
       (- (+ t_1 (/ (* t_3 (fma v -0.25 0.375)) (+ v -1.0))) 4.5)
       t_2))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double t_2 = -1.5 + fma(w, (-0.25 * (r * (r * w))), t_0);
	double t_3 = r * (r * (w * w));
	double t_4 = t_1 + ((t_3 * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_4 <= -2e+16) {
		tmp = (t_1 + ((t_3 * fma(v, -0.25, 0.375)) / (v + -1.0))) - 4.5;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(3.0 + t_0)
	t_2 = Float64(-1.5 + fma(w, Float64(-0.25 * Float64(r * Float64(r * w))), t_0))
	t_3 = Float64(r * Float64(r * Float64(w * w)))
	t_4 = Float64(t_1 + Float64(Float64(t_3 * Float64(0.125 * Float64(Float64(2.0 * v) - 3.0))) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_4 <= -2e+16)
		tmp = Float64(Float64(t_1 + Float64(Float64(t_3 * fma(v, -0.25, 0.375)) / Float64(v + -1.0))) - 4.5);
	else
		tmp = t_2;
	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[(3.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.5 + N[(w * N[(-0.25 * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 + N[(N[(t$95$3 * N[(0.125 * N[(N[(2.0 * v), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$2, If[LessEqual[t$95$4, -2e+16], N[(N[(t$95$1 + N[(N[(t$95$3 * N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$2]]]]]]]

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))) < -inf.0 or -2e16 < (-.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 82.1%

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

   3. Taylor expanded in v around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 81.8

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot 0.25\right)\right) - 4.5 \]

   5. Simplified 81.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot 0.25\right)}\right) - 4.5 \]

   6. Taylor expanded in r around inf

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

   8. Simplified 91.0%

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

   9. Taylor expanded in w around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 95.6

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

   11. Simplified 95.6%

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

   if -inf.0 < (-.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))) < -2e16

   1. Initial program 99.5%

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

   3. Taylor expanded in v around 0

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-fma.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(2 \cdot v - 3\right)\right)}{1 - v} \leq -\infty:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w, -0.25 \cdot \left(r \cdot \left(r \cdot w\right)\right), \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(2 \cdot v - 3\right)\right)}{1 - v} \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w, -0.25 \cdot \left(r \cdot \left(r \cdot w\right)\right), \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := r \cdot \left(r \cdot \left(w \cdot w\right)\right)\\ \mathbf{if}\;\left(3 + t\_0\right) + \frac{t\_1 \cdot \left(0.125 \cdot \left(2 \cdot v - 3\right)\right)}{1 - v} \leq -100:\\ \;\;\;\;-0.25 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* r (* r (* w w)))))
   (if (<=
        (+ (+ 3.0 t_0) (/ (* t_1 (* 0.125 (- (* 2.0 v) 3.0))) (- 1.0 v)))
        -100.0)
     (* -0.25 t_1)
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r * (r * (w * w));
	double tmp;
	if (((3.0 + t_0) + ((t_1 * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -100.0) {
		tmp = -0.25 * t_1;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = r * (r * (w * w))
    if (((3.0d0 + t_0) + ((t_1 * (0.125d0 * ((2.0d0 * v) - 3.0d0))) / (1.0d0 - v))) <= (-100.0d0)) then
        tmp = (-0.25d0) * t_1
    else
        tmp = t_0 + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r * (r * (w * w));
	double tmp;
	if (((3.0 + t_0) + ((t_1 * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -100.0) {
		tmp = -0.25 * t_1;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = r * (r * (w * w))
	tmp = 0
	if ((3.0 + t_0) + ((t_1 * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -100.0:
		tmp = -0.25 * t_1
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(r * Float64(r * Float64(w * w)))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) + Float64(Float64(t_1 * Float64(0.125 * Float64(Float64(2.0 * v) - 3.0))) / Float64(1.0 - v))) <= -100.0)
		tmp = Float64(-0.25 * t_1);
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = r * (r * (w * w));
	tmp = 0.0;
	if (((3.0 + t_0) + ((t_1 * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -100.0)
		tmp = -0.25 * t_1;
	else
		tmp = t_0 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(t$95$1 * N[(0.125 * N[(N[(2.0 * v), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -100.0], N[(-0.25 * t$95$1), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -100

   1. Initial program 79.1%

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

   3. Taylor expanded in v around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 71.6

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot 0.25\right)\right) - 4.5 \]

   5. Simplified 71.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot 0.25\right)}\right) - 4.5 \]

   6. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.2

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

   8. Simplified 77.2%

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

   if -100 < (-.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 87.3%

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

   3. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.3

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

   5. Simplified 82.3%

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

   6. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   7. 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. lower-+.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. lower-/.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. lower-*.f64 94.4

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

   8. Simplified 94.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(2 \cdot v - 3\right)\right)}{1 - v} \leq -100:\\ \;\;\;\;-0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(2 \cdot v - 3\right)\right)}{1 - v} \leq -100:\\ \;\;\;\;w \cdot \left(r \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ 3.0 t_0)
         (/ (* (* r (* r (* w w))) (* 0.125 (- (* 2.0 v) 3.0))) (- 1.0 v)))
        -100.0)
     (* w (* r (* (* r w) -0.375)))
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) + (((r * (r * (w * w))) * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -100.0) {
		tmp = w * (r * ((r * w) * -0.375));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) + (((r * (r * (w * w))) * (0.125d0 * ((2.0d0 * v) - 3.0d0))) / (1.0d0 - v))) <= (-100.0d0)) then
        tmp = w * (r * ((r * w) * (-0.375d0)))
    else
        tmp = t_0 + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) + (((r * (r * (w * w))) * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -100.0) {
		tmp = w * (r * ((r * w) * -0.375));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) + (((r * (r * (w * w))) * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -100.0:
		tmp = w * (r * ((r * w) * -0.375))
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(2.0 * v) - 3.0))) / Float64(1.0 - v))) <= -100.0)
		tmp = Float64(w * Float64(r * Float64(Float64(r * w) * -0.375)));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) + (((r * (r * (w * w))) * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -100.0)
		tmp = w * (r * ((r * w) * -0.375));
	else
		tmp = t_0 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(2.0 * v), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -100.0], N[(w * N[(r * N[(N[(r * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -100

   1. Initial program 79.1%

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

   3. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 68.6

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

   5. Simplified 68.6%

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

   6. Taylor expanded in r around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 71.1

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

   8. Simplified 71.1%

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

   9. Taylor expanded in w around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 72.6

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

   11. Simplified 72.6%

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

   if -100 < (-.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 87.3%

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

   3. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.3

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

   5. Simplified 82.3%

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

   6. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   7. 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. lower-+.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. lower-/.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. lower-*.f64 94.4

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

   8. Simplified 94.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(2 \cdot v - 3\right)\right)}{1 - v} \leq -100:\\ \;\;\;\;w \cdot \left(r \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 77.5

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot 0.25\right)\right) - 4.5 \]

5. Simplified 77.5%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot 0.25\right)}\right) - 4.5 \]

6. Taylor expanded in r around inf

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

8. Simplified 85.9%

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

9. Taylor expanded in w around 0

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

    1. lower-*.f64 N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 91.2

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

11. Simplified 91.2%

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

Alternative 5: 50.8% 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 84.3%

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

   3. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 78.5

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

   5. Simplified 78.5%

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

   6. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 55.8

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

   8. Simplified 55.8%

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

   if 1.1499999999999999 < r

   1. Initial program 81.8%

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

   3. Taylor expanded in v around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 71.6

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot 0.25\right)\right) - 4.5 \]

   5. Simplified 71.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot 0.25\right)}\right) - 4.5 \]

   6. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lft-mult-inverse N/A

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

      15. associate-*l* N/A

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

      16. lower-+.f64 N/A

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

   8. Simplified 29.5%

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

   9. Taylor expanded in r around inf

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

   11. Simplified 29.5%

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

Alternative 6: 57.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 76.2

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

5. Simplified 76.2%

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

6. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
7. 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. lower-+.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. lower-/.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. lower-*.f64 57.2

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

8. Simplified 57.2%

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

9. Final simplification 57.2%

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

Alternative 7: 13.5% 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 83.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 77.5

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot 0.25\right)\right) - 4.5 \]

5. Simplified 77.5%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot 0.25\right)}\right) - 4.5 \]

6. Taylor expanded in r around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-commutative N/A

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

   4. unsub-neg N/A

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

   5. div-sub N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. *-inverses N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. lft-mult-inverse N/A

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

   15. associate-*l* N/A

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

   16. lower-+.f64 N/A

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

8. Simplified 57.2%

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

9. Taylor expanded in r around inf

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

11. Simplified 15.4%

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

Reproduce

herbie shell --seed 2024215 
(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))