Rosa's TurbineBenchmark

Percentage Accurate: 84.3% → 99.8%

Time: 6.6s

Alternatives: 12

Speedup: 1.4×

• 52.8% 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: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

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

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac2 N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-neg.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right), \left(w \cdot r\right) \cdot \frac{w \cdot r}{v - 1}, \mathsf{fma}\left({r}^{-2}, 2, 3\right)\right) - 4.5 \]
8. Add Preprocessing

Alternative 2: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(-0.25 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(r, \left(-0.375 \cdot w\right) \cdot \left(w \cdot r\right), -1.5\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ t_0 3.0)
          (/ (* (* (* (* w w) r) r) (* (- 3.0 (* 2.0 v)) 0.125)) (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (* (* -0.25 (* w r)) (* w r))
     (if (<= t_1 1e+14)
       (+ (fma r (* (* -0.375 w) (* w r)) -1.5) t_0)
       (/ (/ 2.0 r) r)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-0.25 * (w * r)) * (w * r);
	} else if (t_1 <= 1e+14) {
		tmp = fma(r, ((-0.375 * w) * (w * r)), -1.5) + t_0;
	} else {
		tmp = (2.0 / r) / r;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-0.25 * Float64(w * r)) * Float64(w * r));
	elseif (t_1 <= 1e+14)
		tmp = Float64(fma(r, Float64(Float64(-0.375 * w) * Float64(w * r)), -1.5) + t_0);
	else
		tmp = Float64(Float64(2.0 / r) / r);
	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[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.25 * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+14], N[(N[(r * N[(N[(-0.375 * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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

   1. Initial program 81.3%

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 93.4%

       \[\leadsto \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot -0.25\right) \cdot \color{blue}{w} \]
   11. Step-by-step derivation

   12. Applied rewrites 96.6%

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

   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))) < 1e14

   1. Initial program 92.6%

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

   3. Taylor expanded in v around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

   5. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w, -0.125 \cdot v - 0.375, -1.5\right) + \frac{2}{r \cdot r}} \]

   6. Taylor expanded in v around 0

      \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w, \frac{-3}{8}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.1%

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

   10. Applied rewrites 87.8%

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

   if 1e14 < (-.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 82.7%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(-0.25 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(r, \left(-0.375 \cdot w\right) \cdot \left(w \cdot r\right), -1.5\right) + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(-0.25 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.125, v, -0.375\right) \cdot w\right) \cdot w\right) \cdot \left(r \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0
         (-
          (+ (/ 2.0 (* r r)) 3.0)
          (/ (* (* (* (* w w) r) r) (* (- 3.0 (* 2.0 v)) 0.125)) (- 1.0 v)))))
   (if (<= t_0 (- INFINITY))
     (* (* -0.25 (* w r)) (* w r))
     (if (<= t_0 -5e+40)
       (* (* (* (fma -0.125 v -0.375) w) w) (* r r))
       (- (/ (/ 2.0 r) r) 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = ((2.0 / (r * r)) + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-0.25 * (w * r)) * (w * r);
	} else if (t_0 <= -5e+40) {
		tmp = ((fma(-0.125, v, -0.375) * w) * w) * (r * r);
	} else {
		tmp = ((2.0 / r) / r) - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(-0.25 * Float64(w * r)) * Float64(w * r));
	elseif (t_0 <= -5e+40)
		tmp = Float64(Float64(Float64(fma(-0.125, v, -0.375) * w) * w) * Float64(r * r));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.25 * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e+40], N[(N[(N[(N[(-0.125 * v + -0.375), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] * N[(r * r), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] - 1.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 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

   1. Initial program 81.3%

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 93.4%

       \[\leadsto \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot -0.25\right) \cdot \color{blue}{w} \]
   11. Step-by-step derivation

   12. Applied rewrites 96.6%

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

   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))) < -5.00000000000000003e40

   1. Initial program 99.3%

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

   3. Taylor expanded in v around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

   5. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w, -0.125 \cdot v - 0.375, -1.5\right) + \frac{2}{r \cdot r}} \]

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 55.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(-0.125, v, -0.375\right) \cdot w\right) \cdot w\right) \cdot \color{blue}{\left(r \cdot r\right)} \]

   if -5.00000000000000003e40 < (-.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 83.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(-0.25 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.125, v, -0.375\right) \cdot w\right) \cdot w\right) \cdot \left(r \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -((double) INFINITY)) {
		tmp = (-0.25 * (w * r)) * (w * r);
	} else {
		tmp = fma((w * r), ((-0.375 * w) * r), -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(t_0 + 3.0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v))) <= Float64(-Inf))
		tmp = Float64(Float64(-0.25 * Float64(w * r)) * Float64(w * r));
	else
		tmp = Float64(fma(Float64(w * r), Float64(Float64(-0.375 * w) * r), -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[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(-0.25 * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision], N[(N[(N[(w * r), $MachinePrecision] * N[(N[(-0.375 * w), $MachinePrecision] * r), $MachinePrecision] + -1.5), $MachinePrecision] + 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))) < -inf.0

   1. Initial program 81.3%

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 93.4%

       \[\leadsto \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot -0.25\right) \cdot \color{blue}{w} \]
   11. Step-by-step derivation

   12. Applied rewrites 96.6%

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

   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)))

   1. Initial program 85.8%

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

   3. Taylor expanded in v around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

   5. Applied rewrites 83.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w, -0.125 \cdot v - 0.375, -1.5\right) + \frac{2}{r \cdot r}} \]

   6. Taylor expanded in v around 0

      \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w, \frac{-3}{8}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.5%

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

   10. Applied rewrites 96.0%

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

4. Final simplification 96.2%

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

Alternative 5: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\left(-0.25 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (-
       (+ (/ 2.0 (* r r)) 3.0)
       (/ (* (* (* (* w w) r) r) (* (- 3.0 (* 2.0 v)) 0.125)) (- 1.0 v)))
      -5e+40)
   (* (* -0.25 (* w r)) (* w r))
   (- (/ (/ 2.0 r) r) 1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if ((((2.0 / (r * r)) + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -5e+40) {
		tmp = (-0.25 * (w * r)) * (w * r);
	} else {
		tmp = ((2.0 / r) / r) - 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 ((((2.0d0 / (r * r)) + 3.0d0) - (((((w * w) * r) * r) * ((3.0d0 - (2.0d0 * v)) * 0.125d0)) / (1.0d0 - v))) <= (-5d+40)) then
        tmp = ((-0.25d0) * (w * r)) * (w * r)
    else
        tmp = ((2.0d0 / r) / r) - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((((2.0 / (r * r)) + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -5e+40) {
		tmp = (-0.25 * (w * r)) * (w * r);
	} else {
		tmp = ((2.0 / r) / r) - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (((2.0 / (r * r)) + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -5e+40:
		tmp = (-0.25 * (w * r)) * (w * r)
	else:
		tmp = ((2.0 / r) / r) - 1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v))) <= -5e+40)
		tmp = Float64(Float64(-0.25 * Float64(w * r)) * Float64(w * r));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((((2.0 / (r * r)) + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -5e+40)
		tmp = (-0.25 * (w * r)) * (w * r);
	else
		tmp = ((2.0 / r) / r) - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+40], N[(N[(-0.25 * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] - 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))) < -5.00000000000000003e40

   1. Initial program 85.3%

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

   3. Taylor expanded in v around inf

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 79.7%

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

   7. Applied rewrites 82.7%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 80.8%

       \[\leadsto \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot -0.25\right) \cdot \color{blue}{w} \]
   11. Step-by-step derivation

   12. Applied rewrites 83.5%

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

   if -5.00000000000000003e40 < (-.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 83.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\left(-0.25 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\left(-0.25 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ t_0 3.0)
         (/ (* (* (* (* w w) r) r) (* (- 3.0 (* 2.0 v)) 0.125)) (- 1.0 v)))
        -5e+40)
     (* (* -0.25 (* w r)) (* w r))
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -5e+40) {
		tmp = (-0.25 * (w * r)) * (w * r);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) - (((((w * w) * r) * r) * ((3.0d0 - (2.0d0 * v)) * 0.125d0)) / (1.0d0 - v))) <= (-5d+40)) then
        tmp = ((-0.25d0) * (w * r)) * (w * r)
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -5e+40) {
		tmp = (-0.25 * (w * r)) * (w * r);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -5e+40:
		tmp = (-0.25 * (w * r)) * (w * r)
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v))) <= -5e+40)
		tmp = Float64(Float64(-0.25 * Float64(w * r)) * Float64(w * r));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -5e+40)
		tmp = (-0.25 * (w * r)) * (w * r);
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+40], N[(N[(-0.25 * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $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))) < -5.00000000000000003e40

   1. Initial program 85.3%

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

   3. Taylor expanded in v around inf

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 79.7%

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

   7. Applied rewrites 82.7%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 80.8%

       \[\leadsto \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot -0.25\right) \cdot \color{blue}{w} \]
   11. Step-by-step derivation

   12. Applied rewrites 83.5%

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

   if -5.00000000000000003e40 < (-.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 83.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\left(-0.25 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{2}{r \cdot r} + 3\right) - \mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(\frac{r}{1 - v} \cdot w\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5 \end{array} \]

FPCore

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

C

double code(double v, double w, double r) {
	return (((2.0 / (r * r)) + 3.0) - (fma(-0.25, v, 0.375) * (((r / (1.0 - v)) * w) * (w * r)))) - 4.5;
}

Julia

function code(v, w, r)
	return Float64(Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(fma(-0.25, v, 0.375) * Float64(Float64(Float64(r / Float64(1.0 - v)) * w) * Float64(w * r)))) - 4.5)
end

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 90.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 94.4

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 94.4

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 94.4

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

   16. lift-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 94.4

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 94.4

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

6. Applied rewrites 94.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 99.4

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-fma.f64 N/A

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

   19. +-commutative N/A

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

   20. *-commutative N/A

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

   21. distribute-lft-in N/A

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 8: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 720000:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot \left(w \cdot r\right), w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(w \cdot r\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 720000.0)
   (fma (* (* -0.25 r) (* w r)) w (- (/ 2.0 (* r r)) 1.5))
   (-
    (- 3.0 (* (* (* (* (fma v -2.0 3.0) 0.125) w) (/ r (- 1.0 v))) (* w r)))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 720000.0) {
		tmp = fma(((-0.25 * r) * (w * r)), w, ((2.0 / (r * r)) - 1.5));
	} else {
		tmp = (3.0 - ((((fma(v, -2.0, 3.0) * 0.125) * w) * (r / (1.0 - v))) * (w * r))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 720000.0)
		tmp = fma(Float64(Float64(-0.25 * r) * Float64(w * r)), w, Float64(Float64(2.0 / Float64(r * r)) - 1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * w) * Float64(r / Float64(1.0 - v))) * Float64(w * r))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 720000.0], N[(N[(N[(-0.25 * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * w + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * w), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 7.2e5

   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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 93.2%

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

   if 7.2e5 < r

   1. Initial program 86.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.8

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 99.8

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 99.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 720000:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot \left(w \cdot r\right), w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(w \cdot r\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 720000:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot \left(w \cdot r\right), w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(w \cdot \left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right)\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 720000.0)
   (fma (* (* -0.25 r) (* w r)) w (- (/ 2.0 (* r r)) 1.5))
   (-
    (- 3.0 (* (* (* w (* 0.125 (fma -2.0 v 3.0))) (* w r)) (/ r (- 1.0 v))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 720000.0) {
		tmp = fma(((-0.25 * r) * (w * r)), w, ((2.0 / (r * r)) - 1.5));
	} else {
		tmp = (3.0 - (((w * (0.125 * fma(-2.0, v, 3.0))) * (w * r)) * (r / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 720000.0)
		tmp = fma(Float64(Float64(-0.25 * r) * Float64(w * r)), w, Float64(Float64(2.0 / Float64(r * r)) - 1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(w * Float64(0.125 * fma(-2.0, v, 3.0))) * Float64(w * r)) * Float64(r / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 720000.0], N[(N[(N[(-0.25 * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * w + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(w * N[(0.125 * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 7.2e5

   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

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 93.2%

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

   if 7.2e5 < r

   1. Initial program 86.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 94.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 720000:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot \left(w \cdot r\right), w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(w \cdot \left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right)\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot \left(w \cdot r\right), w, t\_0 - 1.5\right)\\ \mathbf{if}\;v \leq -2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(w \cdot r, \left(-0.375 \cdot w\right) \cdot r, -1.5\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (fma (* (* -0.25 r) (* w r)) w (- t_0 1.5))))
   (if (<= v -2.0)
     t_1
     (if (<= v 4e-68) (+ (fma (* w r) (* (* -0.375 w) r) -1.5) t_0) t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = fma(((-0.25 * r) * (w * r)), w, (t_0 - 1.5));
	double tmp;
	if (v <= -2.0) {
		tmp = t_1;
	} else if (v <= 4e-68) {
		tmp = fma((w * r), ((-0.375 * w) * r), -1.5) + t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = fma(Float64(Float64(-0.25 * r) * Float64(w * r)), w, Float64(t_0 - 1.5))
	tmp = 0.0
	if (v <= -2.0)
		tmp = t_1;
	elseif (v <= 4e-68)
		tmp = Float64(fma(Float64(w * r), Float64(Float64(-0.375 * w) * r), -1.5) + t_0);
	else
		tmp = 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[(N[(N[(-0.25 * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * w + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2.0], t$95$1, If[LessEqual[v, 4e-68], N[(N[(N[(w * r), $MachinePrecision] * N[(N[(-0.375 * w), $MachinePrecision] * r), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2 or 4.00000000000000027e-68 < 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 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. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 97.1%

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

   if -2 < v < 4.00000000000000027e-68

   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 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

   5. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w, -0.125 \cdot v - 0.375, -1.5\right) + \frac{2}{r \cdot r}} \]

   6. Taylor expanded in v around 0

      \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w, \frac{-3}{8}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.4%

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

   10. Applied rewrites 99.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot \left(w \cdot r\right), w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(w \cdot r, \left(-0.375 \cdot w\right) \cdot r, -1.5\right) + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot \left(w \cdot r\right), w, \frac{2}{r \cdot r} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.1% 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 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 w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 57.9

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

5. Applied rewrites 57.9%

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

Alternative 12: 43.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return 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 = 2.0d0 / (r * r)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 r around 0

   \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
4. 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 47.8

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

5. Applied rewrites 47.8%

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

Reproduce

herbie shell --seed 2024332 
(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))