Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 99.8%

Time: 13.9s

Alternatives: 8

Speedup: 1.7×

• 53.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 10^{+64}:\\ \;\;\;\;t\_0 + \left(-1.5 + \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 1e+64)
     (+
      t_0
      (+
       -1.5
       (* (fma v -2.0 3.0) (/ (* 0.125 (* r (* w (* r w)))) (+ v -1.0)))))
     (+
      t_0
      (+
       -1.5
       (* (fma v -2.0 3.0) (/ (* 0.125 (* w (* r (* r w)))) (+ v -1.0))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 1e+64) {
		tmp = t_0 + (-1.5 + (fma(v, -2.0, 3.0) * ((0.125 * (r * (w * (r * w)))) / (v + -1.0))));
	} else {
		tmp = t_0 + (-1.5 + (fma(v, -2.0, 3.0) * ((0.125 * (w * (r * (r * w)))) / (v + -1.0))));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 1e+64)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(fma(v, -2.0, 3.0) * Float64(Float64(0.125 * Float64(r * Float64(w * Float64(r * w)))) / Float64(v + -1.0)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(fma(v, -2.0, 3.0) * Float64(Float64(0.125 * Float64(w * Float64(r * Float64(r * w)))) / Float64(v + -1.0)))));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 1e+64], N[(t$95$0 + N[(-1.5 + N[(N[(v * -2.0 + 3.0), $MachinePrecision] * N[(N[(0.125 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(v * -2.0 + 3.0), $MachinePrecision] * N[(N[(0.125 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.00000000000000002e64

   1. Initial program 92.2%

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

   3. Simplified 95.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right)}{v + -1} + -1.5\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right)}{v + -1} + -1.5\right) \]

   if 1.00000000000000002e64 < (*.f64 w w)

   1. Initial program 81.7%

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

   3. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 94.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right)}{v + -1} + -1.5\right) \]

      2. associate-*r* 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \color{blue}{\left(\left(r \cdot \left(r \cdot w\right)\right) \cdot w\right)}}{v + -1} + -1.5\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \color{blue}{\left(\left(r \cdot \left(r \cdot w\right)\right) \cdot w\right)}}{v + -1} + -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+64}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (sqrt 0.125))))
   (+
    (/ 2.0 (* r r))
    (- -1.5 (* (fma v -2.0 3.0) (* t_0 (/ t_0 (- 1.0 v))))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * sqrt(0.125);
	return (2.0 / (r * r)) + (-1.5 - (fma(v, -2.0, 3.0) * (t_0 * (t_0 / (1.0 - v)))));
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * sqrt(0.125))
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(fma(v, -2.0, 3.0) * Float64(t_0 * Float64(t_0 / Float64(1.0 - v))))))
end

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Simplified 89.3%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} + -1.5\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 89.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\color{blue}{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)} \cdot \sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}}{v + -1} + -1.5\right) \]

   2. associate-/l* 89.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \color{blue}{\left(\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)} \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right)} + -1.5\right) \]

   3. *-commutative 89.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\sqrt{\color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot 0.125}} \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

   4. *-commutative 89.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\sqrt{\color{blue}{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right)} \cdot 0.125} \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

   5. *-commutative 89.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\sqrt{\left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right) \cdot 0.125} \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

   6. sqrt-prod 89.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\color{blue}{\left(\sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r} \cdot \sqrt{0.125}\right)} \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

   7. associate-*l* 83.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\left(\sqrt{\color{blue}{\left(w \cdot w\right) \cdot \left(r \cdot r\right)}} \cdot \sqrt{0.125}\right) \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

   8. sqrt-prod 83.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\left(\color{blue}{\left(\sqrt{w \cdot w} \cdot \sqrt{r \cdot r}\right)} \cdot \sqrt{0.125}\right) \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

   9. sqrt-prod 43.0%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\left(\left(\color{blue}{\left(\sqrt{w} \cdot \sqrt{w}\right)} \cdot \sqrt{r \cdot r}\right) \cdot \sqrt{0.125}\right) \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

   10. add-sqr-sqrt 65.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\left(\left(\color{blue}{w} \cdot \sqrt{r \cdot r}\right) \cdot \sqrt{0.125}\right) \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

   11. sqrt-unprod 38.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\left(\left(w \cdot \color{blue}{\left(\sqrt{r} \cdot \sqrt{r}\right)}\right) \cdot \sqrt{0.125}\right) \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

   12. add-sqr-sqrt 72.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\left(\left(w \cdot \color{blue}{r}\right) \cdot \sqrt{0.125}\right) \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

   13. *-commutative 72.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \left(\left(\color{blue}{\left(r \cdot w\right)} \cdot \sqrt{0.125}\right) \cdot \frac{\sqrt{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1}\right) + -1.5\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \color{blue}{\left(\left(\left(r \cdot w\right) \cdot \sqrt{0.125}\right) \cdot \frac{\left(r \cdot w\right) \cdot \sqrt{0.125}}{v + -1}\right)} + -1.5\right) \]

7. Final simplification 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \mathsf{fma}\left(v, -2, 3\right) \cdot \left(\left(\left(r \cdot w\right) \cdot \sqrt{0.125}\right) \cdot \frac{\left(r \cdot w\right) \cdot \sqrt{0.125}}{1 - v}\right)\right) \]
8. Add Preprocessing

Alternative 3: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 + \frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot {\left(r \cdot w\right)}^{2}}}\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (+ -1.5 (/ (fma v -2.0 3.0) (/ (+ v -1.0) (* 0.125 (pow (* r w) 2.0)))))))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + (fma(v, -2.0, 3.0) / ((v + -1.0) / (0.125 * pow((r * w), 2.0)))));
}

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(fma(v, -2.0, 3.0) / Float64(Float64(v + -1.0) / Float64(0.125 * (Float64(r * w) ^ 2.0))))))
end

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Simplified 89.3%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} + -1.5\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 89.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \color{blue}{\frac{1}{\frac{v + -1}{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}} + -1.5\right) \]

   2. un-div-inv 89.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}} + -1.5\right) \]

   3. *-commutative 89.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot \color{blue}{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right)}}} + -1.5\right) \]

   4. *-commutative 89.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}} + -1.5\right) \]

   5. add-sqr-sqrt 89.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot \color{blue}{\left(\sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r} \cdot \sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}\right)}}} + -1.5\right) \]

   6. pow2 89.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot \color{blue}{{\left(\sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}\right)}^{2}}}} + -1.5\right) \]

   7. associate-*l* 83.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot {\left(\sqrt{\color{blue}{\left(w \cdot w\right) \cdot \left(r \cdot r\right)}}\right)}^{2}}} + -1.5\right) \]

   8. sqrt-prod 83.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot {\color{blue}{\left(\sqrt{w \cdot w} \cdot \sqrt{r \cdot r}\right)}}^{2}}} + -1.5\right) \]

   9. sqrt-prod 47.0%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot {\left(\color{blue}{\left(\sqrt{w} \cdot \sqrt{w}\right)} \cdot \sqrt{r \cdot r}\right)}^{2}}} + -1.5\right) \]

   10. add-sqr-sqrt 91.7%

       \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot {\left(\color{blue}{w} \cdot \sqrt{r \cdot r}\right)}^{2}}} + -1.5\right) \]

   11. sqrt-unprod 53.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot {\left(w \cdot \color{blue}{\left(\sqrt{r} \cdot \sqrt{r}\right)}\right)}^{2}}} + -1.5\right) \]

   12. add-sqr-sqrt 99.6%

       \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot {\left(w \cdot \color{blue}{r}\right)}^{2}}} + -1.5\right) \]

   13. *-commutative 99.6%

       \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot {\color{blue}{\left(r \cdot w\right)}}^{2}}} + -1.5\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot {\left(r \cdot w\right)}^{2}}}} + -1.5\right) \]

7. Final simplification 99.6%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 + \frac{\mathsf{fma}\left(v, -2, 3\right)}{\frac{v + -1}{0.125 \cdot {\left(r \cdot w\right)}^{2}}}\right) \]
8. Add Preprocessing

Alternative 4: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 10^{+186}:\\ \;\;\;\;t\_0 + \left(-1.5 + \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + {\left(r \cdot w\right)}^{2} \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 1e+186)
     (+
      t_0
      (+
       -1.5
       (* (fma v -2.0 3.0) (/ (* 0.125 (* r (* w (* r w)))) (+ v -1.0)))))
     (+ t_0 (+ -1.5 (* (pow (* r w) 2.0) -0.25))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 1e+186) {
		tmp = t_0 + (-1.5 + (fma(v, -2.0, 3.0) * ((0.125 * (r * (w * (r * w)))) / (v + -1.0))));
	} else {
		tmp = t_0 + (-1.5 + (pow((r * w), 2.0) * -0.25));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (w <= 1e+186)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(fma(v, -2.0, 3.0) * Float64(Float64(0.125 * Float64(r * Float64(w * Float64(r * w)))) / Float64(v + -1.0)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64((Float64(r * w) ^ 2.0) * -0.25)));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 1e+186], N[(t$95$0 + N[(-1.5 + N[(N[(v * -2.0 + 3.0), $MachinePrecision] * N[(N[(0.125 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 9.9999999999999998e185

   1. Initial program 89.6%

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

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 98.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right)}{v + -1} + -1.5\right) \]

   6. Applied egg-rr 98.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right)}{v + -1} + -1.5\right) \]

   if 9.9999999999999998e185 < w

   1. Initial program 69.6%

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

   3. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 69.6%

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

      1. unpow2 69.6%

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

      2. unpow2 69.6%

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

      3. swap-sqr 99.9%

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

      4. unpow2 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 10^{+186}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + {\left(r \cdot w\right)}^{2} \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1 \cdot 10^{+110} \lor \neg \left(v \leq 10^{-56}\right):\\ \;\;\;\;t\_0 + \left(-1.5 + {\left(r \cdot w\right)}^{2} \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) + \frac{\left(r \cdot w\right) \cdot \left(0.125 \cdot \left(r \cdot \left(w \cdot \left(3 + v \cdot -2\right)\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1e+110) (not (<= v 1e-56)))
     (+ t_0 (+ -1.5 (* (pow (* r w) 2.0) -0.25)))
     (-
      (+
       (+ t_0 3.0)
       (/ (* (* r w) (* 0.125 (* r (* w (+ 3.0 (* v -2.0)))))) (+ v -1.0)))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1e+110) || !(v <= 1e-56)) {
		tmp = t_0 + (-1.5 + (pow((r * w), 2.0) * -0.25));
	} else {
		tmp = ((t_0 + 3.0) + (((r * w) * (0.125 * (r * (w * (3.0 + (v * -2.0)))))) / (v + -1.0))) - 4.5;
	}
	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 ((v <= (-1d+110)) .or. (.not. (v <= 1d-56))) then
        tmp = t_0 + ((-1.5d0) + (((r * w) ** 2.0d0) * (-0.25d0)))
    else
        tmp = ((t_0 + 3.0d0) + (((r * w) * (0.125d0 * (r * (w * (3.0d0 + (v * (-2.0d0))))))) / (v + (-1.0d0)))) - 4.5d0
    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 ((v <= -1e+110) || !(v <= 1e-56)) {
		tmp = t_0 + (-1.5 + (Math.pow((r * w), 2.0) * -0.25));
	} else {
		tmp = ((t_0 + 3.0) + (((r * w) * (0.125 * (r * (w * (3.0 + (v * -2.0)))))) / (v + -1.0))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1e+110) or not (v <= 1e-56):
		tmp = t_0 + (-1.5 + (math.pow((r * w), 2.0) * -0.25))
	else:
		tmp = ((t_0 + 3.0) + (((r * w) * (0.125 * (r * (w * (3.0 + (v * -2.0)))))) / (v + -1.0))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1e+110) || !(v <= 1e-56))
		tmp = Float64(t_0 + Float64(-1.5 + Float64((Float64(r * w) ^ 2.0) * -0.25)));
	else
		tmp = Float64(Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * w) * Float64(0.125 * Float64(r * Float64(w * Float64(3.0 + Float64(v * -2.0)))))) / Float64(v + -1.0))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -1e+110) || ~((v <= 1e-56)))
		tmp = t_0 + (-1.5 + (((r * w) ^ 2.0) * -0.25));
	else
		tmp = ((t_0 + 3.0) + (((r * w) * (0.125 * (r * (w * (3.0 + (v * -2.0)))))) / (v + -1.0))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1e+110], N[Not[LessEqual[v, 1e-56]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(0.125 * N[(r * N[(w * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1e110 or 1e-56 < v

   1. Initial program 84.4%

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

   3. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 80.3%

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

      1. unpow2 80.3%

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

      2. unpow2 80.3%

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

      3. swap-sqr 99.9%

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

      4. unpow2 99.9%

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

   7. Simplified 99.9%

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

   if -1e110 < v < 1e-56

   1. Initial program 90.0%

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

      1. add-sqr-sqrt 89.9%

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

      2. associate-*r* 89.9%

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

      3. associate-*l* 85.8%

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

      4. sqrt-prod 85.8%

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

      5. sqrt-prod 48.2%

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

      6. add-sqr-sqrt 69.6%

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

      7. sqrt-unprod 37.6%

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

      8. add-sqr-sqrt 71.0%

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

      9. *-commutative 71.0%

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

      10. sub-neg 71.0%

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

      11. +-commutative 71.0%

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

      12. *-commutative 71.0%

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

      13. distribute-rgt-neg-in 71.0%

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

      14. metadata-eval 71.0%

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

      15. fma-undefine 71.0%

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

      16. associate-*l* 66.0%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in r around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+110} \lor \neg \left(v \leq 10^{-56}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + {\left(r \cdot w\right)}^{2} \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot w\right) \cdot \left(0.125 \cdot \left(r \cdot \left(w \cdot \left(3 + v \cdot -2\right)\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 10^{-42}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) + \frac{\left(r \cdot w\right) \cdot \left(0.125 \cdot \left(r \cdot \left(w \cdot \left(3 + v \cdot -2\right)\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 1e-42)
     (+ t_0 (+ -1.5 (* (* r w) (* r (* w -0.375)))))
     (-
      (+
       (+ t_0 3.0)
       (/ (* (* r w) (* 0.125 (* r (* w (+ 3.0 (* v -2.0)))))) (+ v -1.0)))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1e-42) {
		tmp = t_0 + (-1.5 + ((r * w) * (r * (w * -0.375))));
	} else {
		tmp = ((t_0 + 3.0) + (((r * w) * (0.125 * (r * (w * (3.0 + (v * -2.0)))))) / (v + -1.0))) - 4.5;
	}
	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 (r <= 1d-42) then
        tmp = t_0 + ((-1.5d0) + ((r * w) * (r * (w * (-0.375d0)))))
    else
        tmp = ((t_0 + 3.0d0) + (((r * w) * (0.125d0 * (r * (w * (3.0d0 + (v * (-2.0d0))))))) / (v + (-1.0d0)))) - 4.5d0
    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 (r <= 1e-42) {
		tmp = t_0 + (-1.5 + ((r * w) * (r * (w * -0.375))));
	} else {
		tmp = ((t_0 + 3.0) + (((r * w) * (0.125 * (r * (w * (3.0 + (v * -2.0)))))) / (v + -1.0))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 1e-42:
		tmp = t_0 + (-1.5 + ((r * w) * (r * (w * -0.375))))
	else:
		tmp = ((t_0 + 3.0) + (((r * w) * (0.125 * (r * (w * (3.0 + (v * -2.0)))))) / (v + -1.0))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 1e-42)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(r * Float64(w * -0.375)))));
	else
		tmp = Float64(Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * w) * Float64(0.125 * Float64(r * Float64(w * Float64(3.0 + Float64(v * -2.0)))))) / Float64(v + -1.0))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 1e-42)
		tmp = t_0 + (-1.5 + ((r * w) * (r * (w * -0.375))));
	else
		tmp = ((t_0 + 3.0) + (((r * w) * (0.125 * (r * (w * (3.0 + (v * -2.0)))))) / (v + -1.0))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1e-42], N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(0.125 * N[(r * N[(w * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.00000000000000004e-42

   1. Initial program 85.8%

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

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 83.4%

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

      1. unpow2 83.4%

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

      2. unpow2 83.4%

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

      3. swap-sqr 95.0%

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

      4. unpow2 95.0%

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

   7. Simplified 95.0%

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

      1. unpow2 95.0%

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

   9. Applied egg-rr 95.0%

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

       1. associate-*r* 95.0%

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

   11. Applied egg-rr 95.0%

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

       1. *-commutative 95.0%

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

   13. Applied egg-rr 95.0%

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

       1. associate-*l* 95.0%

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

   15. Simplified 95.0%

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

   if 1.00000000000000004e-42 < r

   1. Initial program 91.7%

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

      1. add-sqr-sqrt 91.6%

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

      2. associate-*r* 91.6%

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

      3. associate-*l* 80.1%

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

      4. sqrt-prod 80.1%

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

      5. sqrt-prod 48.2%

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

      6. add-sqr-sqrt 50.8%

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

      7. sqrt-unprod 60.1%

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

      8. add-sqr-sqrt 60.1%

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

      9. *-commutative 60.1%

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

      10. sub-neg 60.1%

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

      11. +-commutative 60.1%

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

      12. *-commutative 60.1%

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

      13. distribute-rgt-neg-in 60.1%

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

      14. metadata-eval 60.1%

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

      15. fma-undefine 60.1%

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

      16. associate-*l* 50.8%

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

   4. Applied egg-rr 96.2%

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

   5. Taylor expanded in r around 0 96.3%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 10^{-42}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot w\right) \cdot \left(0.125 \cdot \left(r \cdot \left(w \cdot \left(3 + v \cdot -2\right)\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+ (/ 2.0 (* r r)) (+ -1.5 (* (* r w) (* r (* w -0.375))))))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((r * w) * (r * (w * -0.375))));
}

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) + ((r * w) * (r * (w * (-0.375d0)))))
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((r * w) * (r * (w * -0.375))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((r * w) * (r * (w * -0.375))))

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(r * w) * Float64(r * Float64(w * -0.375)))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + ((r * w) * (r * (w * -0.375))));
end

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.6%

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

3. Simplified 89.3%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} + -1.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 82.0%

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

   1. unpow2 82.0%

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

   2. unpow2 82.0%

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

   3. swap-sqr 94.7%

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

   4. unpow2 94.7%

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

7. Simplified 94.7%

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

   1. unpow2 94.7%

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

9. Applied egg-rr 94.7%

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

    1. associate-*r* 94.7%

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

11. Applied egg-rr 94.7%

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

    1. *-commutative 94.7%

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

13. Applied egg-rr 94.7%

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

    1. associate-*l* 94.7%

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

15. Simplified 94.7%

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

16. Final simplification 94.7%

    \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\right) \]
17. Add Preprocessing

Alternative 8: 93.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+ (/ 2.0 (* r r)) (+ -1.5 (* -0.375 (* (* r w) (* r w))))))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + (-0.375 * ((r * w) * (r * w))));
}

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) + ((-0.375d0) * ((r * w) * (r * w))))
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + (-0.375 * ((r * w) * (r * w))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + (-0.375 * ((r * w) * (r * w))))

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(-0.375 * Float64(Float64(r * w) * Float64(r * w)))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + (-0.375 * ((r * w) * (r * w))));
end

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(-0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.6%

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

3. Simplified 89.3%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} + -1.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 82.0%

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

   1. unpow2 82.0%

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

   2. unpow2 82.0%

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

   3. swap-sqr 94.7%

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

   4. unpow2 94.7%

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

7. Simplified 94.7%

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

   1. unpow2 94.7%

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

9. Applied egg-rr 94.7%

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

10. Final simplification 94.7%

    \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))