Rosa's TurbineBenchmark

Percentage Accurate: 84.7% → 99.7%

Time: 11.7s

Alternatives: 12

Speedup: 1.4×

• 53.1% 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: 84.7% 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: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.3%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.8

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(-0.25 * w) * r) * Float64(w * r)) + t_0);
	elseif (t_1 <= 3.0)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * w) * r) * Float64(w * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = fma(-0.375, Float64(Float64(w * Float64(r * r)) * w), Float64(t_0 - 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(-0.25 * w), $MachinePrecision] * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 3.0], N[(N[(3.0 - N[(N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(-0.375 * N[(N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] + N[(t$95$0 - 1.5), $MachinePrecision]), $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 78.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. 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. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 92.7

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

   7. Applied rewrites 92.7%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 92.7%

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

   12. Applied rewrites 96.5%

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

   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))) < 3

   1. Initial program 88.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.7

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

   4. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 99.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.7

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 99.7%

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

   if 3 < (-.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 85.1%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 98.5%

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

Alternative 3: 95.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(3 + t\_0\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot w\right) \cdot r\right) \cdot \left(w \cdot r\right) + t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot r\right) \cdot w\right) \cdot r, t\_0 - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ 3.0 t_0)
          (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (+ (* (* (* -0.25 w) r) (* w r)) t_0)
     (if (<= t_1 2e+307)
       (fma -0.375 (* (* (* w r) w) r) (- t_0 1.5))
       (/ (/ 2.0 r) r)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((-0.25 * w) * r) * (w * r)) + t_0;
	} else if (t_1 <= 2e+307) {
		tmp = fma(-0.375, (((w * r) * w) * r), (t_0 - 1.5));
	} 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(3.0 + t_0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(-0.25 * w) * r) * Float64(w * r)) + t_0);
	elseif (t_1 <= 2e+307)
		tmp = fma(-0.375, Float64(Float64(Float64(w * r) * w) * r), Float64(t_0 - 1.5));
	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[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(-0.25 * w), $MachinePrecision] * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e+307], N[(-0.375 * N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] + N[(t$95$0 - 1.5), $MachinePrecision]), $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 78.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. 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. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 92.7

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

   7. Applied rewrites 92.7%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 92.7%

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

   12. Applied rewrites 96.5%

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

   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))) < 1.99999999999999997e307

   1. Initial program 93.6%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 89.5%

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

   7. Applied rewrites 94.1%

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

   if 1.99999999999999997e307 < (-.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 72.2%

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

   3. Taylor expanded in 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 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot w\right) \cdot r\right) \cdot \left(w \cdot r\right) + \frac{2}{r \cdot r}\\ \mathbf{elif}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot r\right) \cdot w\right) \cdot r, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+299}:\\ \;\;\;\;\left(\left(-0.25 \cdot w\right) \cdot r\right) \cdot \left(w \cdot r\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot w, \left(w \cdot r\right) \cdot r, -1.5\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
        -1e+299)
     (+ (* (* (* -0.25 w) r) (* w r)) t_0)
     (+ (fma (* -0.375 w) (* (* w r) r) -1.5) t_0))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+299) {
		tmp = (((-0.25 * w) * r) * (w * r)) + t_0;
	} else {
		tmp = fma((-0.375 * w), ((w * r) * 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(3.0 + t_0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= -1e+299)
		tmp = Float64(Float64(Float64(Float64(-0.25 * w) * r) * Float64(w * r)) + t_0);
	else
		tmp = Float64(fma(Float64(-0.375 * w), Float64(Float64(w * r) * 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[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+299], N[(N[(N[(N[(-0.25 * w), $MachinePrecision] * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(-0.375 * w), $MachinePrecision] * N[(N[(w * r), $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))) < -1.0000000000000001e299

   1. Initial program 78.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. 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. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.8

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

   7. Applied rewrites 91.8%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 91.8%

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

   12. Applied rewrites 95.7%

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

   if -1.0000000000000001e299 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 86.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. 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. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in r around 0

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

   7. Applied rewrites 82.9%

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

   8. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 74.6

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

   10. Applied rewrites 74.6%

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

   12. Applied rewrites 94.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+299}:\\ \;\;\;\;\left(\left(-0.25 \cdot w\right) \cdot r\right) \cdot \left(w \cdot r\right) + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot w, \left(w \cdot r\right) \cdot r, -1.5\right) + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(-0.25 \cdot w\right) \cdot r\right) \cdot \left(w \cdot r\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
        -1e+45)
     (+ (* (* (* -0.25 w) r) (* w r)) t_0)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45) {
		tmp = (((-0.25 * w) * r) * (w * r)) + t_0;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((((w * w) * r) * r) * ((3.0d0 - (v * 2.0d0)) * 0.125d0)) / (1.0d0 - v))) <= (-1d+45)) then
        tmp = ((((-0.25d0) * w) * r) * (w * r)) + t_0
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45) {
		tmp = (((-0.25 * w) * r) * (w * r)) + t_0;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45:
		tmp = (((-0.25 * w) * r) * (w * r)) + t_0
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= -1e+45)
		tmp = Float64(Float64(Float64(Float64(-0.25 * w) * r) * Float64(w * r)) + t_0);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45)
		tmp = (((-0.25 * w) * r) * (w * r)) + t_0;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+45], N[(N[(N[(N[(-0.25 * w), $MachinePrecision] * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -9.9999999999999993e44

   1. Initial program 82.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. 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. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 81.8

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

   7. Applied rewrites 81.8%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 81.8%

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

   12. Applied rewrites 85.6%

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

   if -9.9999999999999993e44 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 84.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(-0.25 \cdot w\right) \cdot r\right) \cdot \left(w \cdot r\right) + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
        -1e+45)
     (+ (* (* (* -0.375 (* r r)) w) w) t_0)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45) {
		tmp = (((-0.375 * (r * r)) * w) * w) + t_0;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((((w * w) * r) * r) * ((3.0d0 - (v * 2.0d0)) * 0.125d0)) / (1.0d0 - v))) <= (-1d+45)) then
        tmp = ((((-0.375d0) * (r * r)) * w) * w) + t_0
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45) {
		tmp = (((-0.375 * (r * r)) * w) * w) + t_0;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45:
		tmp = (((-0.375 * (r * r)) * w) * w) + t_0
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= -1e+45)
		tmp = Float64(Float64(Float64(Float64(-0.375 * Float64(r * r)) * w) * w) + t_0);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45)
		tmp = (((-0.375 * (r * r)) * w) * w) + t_0;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+45], N[(N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -9.9999999999999993e44

   1. Initial program 82.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. 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. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in r around 0

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 74.4

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

   10. Applied rewrites 74.4%

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

   11. Taylor expanded in r around inf

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

   13. Applied rewrites 83.1%

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

   if -9.9999999999999993e44 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 84.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
        -1e+45)
     (* (* (* -0.375 (* r r)) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((((w * w) * r) * r) * ((3.0d0 - (v * 2.0d0)) * 0.125d0)) / (1.0d0 - v))) <= (-1d+45)) then
        tmp = (((-0.375d0) * (r * r)) * w) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45:
		tmp = ((-0.375 * (r * r)) * w) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= -1e+45)
		tmp = Float64(Float64(Float64(-0.375 * Float64(r * r)) * w) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45)
		tmp = ((-0.375 * (r * r)) * w) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+45], N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $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))) < -9.9999999999999993e44

   1. Initial program 82.1%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 77.6%

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

   if -9.9999999999999993e44 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 84.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
        -1e+45)
     (* (* (* -0.25 (* r r)) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((((w * w) * r) * r) * ((3.0d0 - (v * 2.0d0)) * 0.125d0)) / (1.0d0 - v))) <= (-1d+45)) then
        tmp = (((-0.25d0) * (r * r)) * w) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45:
		tmp = ((-0.25 * (r * r)) * w) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= -1e+45)
		tmp = Float64(Float64(Float64(-0.25 * Float64(r * r)) * w) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+45)
		tmp = ((-0.25 * (r * r)) * w) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+45], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $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))) < -9.9999999999999993e44

   1. Initial program 82.1%

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

   3. Taylor expanded in 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 6.8

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

   5. Applied rewrites 6.8%

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

   6. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

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

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. associate-*l/ N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

   8. Applied rewrites 78.1%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 75.9%

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

   if -9.9999999999999993e44 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 84.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 3.5 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(w \cdot \left(r \cdot r\right)\right) \cdot w, t\_0 - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \mathsf{fma}\left(\left(\left(\frac{r}{1 - v} \cdot w\right) \cdot w\right) \cdot r, 0.125 \cdot \mathsf{fma}\left(-2, v, 3\right), 4.5\right)\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 3.5e-65)
     (fma -0.375 (* (* w (* r r)) w) (- t_0 1.5))
     (+
      (-
       3.0
       (fma (* (* (* (/ r (- 1.0 v)) w) w) r) (* 0.125 (fma -2.0 v 3.0)) 4.5))
      t_0))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3.5e-65) {
		tmp = fma(-0.375, ((w * (r * r)) * w), (t_0 - 1.5));
	} else {
		tmp = (3.0 - fma(((((r / (1.0 - v)) * w) * w) * r), (0.125 * fma(-2.0, v, 3.0)), 4.5)) + t_0;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 3.5e-65)
		tmp = fma(-0.375, Float64(Float64(w * Float64(r * r)) * w), Float64(t_0 - 1.5));
	else
		tmp = Float64(Float64(3.0 - fma(Float64(Float64(Float64(Float64(r / Float64(1.0 - v)) * w) * w) * r), Float64(0.125 * fma(-2.0, v, 3.0)), 4.5)) + 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[r, 3.5e-65], N[(-0.375 * N[(N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * N[(0.125 * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.50000000000000005e-65

   1. Initial program 79.1%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 91.5%

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

   if 3.50000000000000005e-65 < r

   1. Initial program 92.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.5 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(w \cdot \left(r \cdot r\right)\right) \cdot w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \mathsf{fma}\left(\left(\left(\frac{r}{1 - v} \cdot w\right) \cdot w\right) \cdot r, 0.125 \cdot \mathsf{fma}\left(-2, v, 3\right), 4.5\right)\right) + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \mathsf{fma}\left(\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot r, w, -1.5\right) + t\_0\\ \mathbf{if}\;v \leq -3.5 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot w, \left(w \cdot r\right) \cdot r, -1.5\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (+ (fma (* (* (* -0.25 r) w) r) w -1.5) t_0)))
   (if (<= v -3.5e+55)
     t_1
     (if (<= v 5e-25) (+ (fma (* -0.375 w) (* (* w r) 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, -1.5) + t_0;
	double tmp;
	if (v <= -3.5e+55) {
		tmp = t_1;
	} else if (v <= 5e-25) {
		tmp = fma((-0.375 * w), ((w * r) * 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 = Float64(fma(Float64(Float64(Float64(-0.25 * r) * w) * r), w, -1.5) + t_0)
	tmp = 0.0
	if (v <= -3.5e+55)
		tmp = t_1;
	elseif (v <= 5e-25)
		tmp = Float64(fma(Float64(-0.375 * w), Float64(Float64(w * r) * 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[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * w + -1.5), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[v, -3.5e+55], t$95$1, If[LessEqual[v, 5e-25], N[(N[(N[(-0.375 * w), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.5000000000000001e55 or 4.99999999999999962e-25 < 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. 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. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 90.9

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

   7. Applied rewrites 90.9%

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

   9. Applied rewrites 96.6%

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

   if -3.5000000000000001e55 < v < 4.99999999999999962e-25

   1. Initial program 83.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. 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. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in r around 0

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 75.7

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

   10. Applied rewrites 75.7%

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

   12. Applied rewrites 97.6%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.5 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot r, w, -1.5\right) + \frac{2}{r \cdot r}\\ \mathbf{elif}\;v \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot w, \left(w \cdot r\right) \cdot r, -1.5\right) + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot r, w, -1.5\right) + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 55.0

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

5. Applied rewrites 55.0%

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

Alternative 12: 44.3% 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 83.3%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 42.6

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

5. Applied rewrites 42.6%

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

Reproduce

herbie shell --seed 2024241 
(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))