Rosa's TurbineBenchmark

Percentage Accurate: 85.1% → 99.7%

Time: 10.0s

Alternatives: 5

Speedup: 1.7×

• 52.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

3. Simplified 88.9%

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

   1. clear-num 89.0%

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

   2. un-div-inv 88.9%

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

   3. associate-*r* 81.5%

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

   4. pow2 81.5%

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

   5. pow2 81.5%

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

   6. pow-prod-down 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -10000000 \lor \neg \left(v \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;t\_0 + \left(-1.5 + -0.25 \cdot \frac{r \cdot w}{\frac{\frac{1}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -10000000.0) (not (<= v 4e-32)))
     (+ t_0 (+ -1.5 (* -0.25 (/ (* r w) (/ (/ 1.0 r) w)))))
     (+ t_0 (- -1.5 (* 0.375 (* (* r w) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -10000000.0) || !(v <= 4e-32)) {
		tmp = t_0 + (-1.5 + (-0.25 * ((r * w) / ((1.0 / r) / w))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	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 <= (-10000000.0d0)) .or. (.not. (v <= 4d-32))) then
        tmp = t_0 + ((-1.5d0) + ((-0.25d0) * ((r * w) / ((1.0d0 / r) / w))))
    else
        tmp = t_0 + ((-1.5d0) - (0.375d0 * ((r * w) * (r * w))))
    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 <= -10000000.0) || !(v <= 4e-32)) {
		tmp = t_0 + (-1.5 + (-0.25 * ((r * w) / ((1.0 / r) / w))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -10000000.0) or not (v <= 4e-32):
		tmp = t_0 + (-1.5 + (-0.25 * ((r * w) / ((1.0 / r) / w))))
	else:
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -10000000.0) || !(v <= 4e-32))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(-0.25 * Float64(Float64(r * w) / Float64(Float64(1.0 / r) / w)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -10000000.0) || ~((v <= 4e-32)))
		tmp = t_0 + (-1.5 + (-0.25 * ((r * w) / ((1.0 / r) / w))));
	else
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	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, -10000000.0], N[Not[LessEqual[v, 4e-32]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(-0.25 * N[(N[(r * w), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1e7 or 4.00000000000000022e-32 < v

   1. Initial program 82.5%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in v around inf 84.7%

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

      1. *-commutative 84.7%

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

      2. unpow2 84.7%

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

      3. unpow2 84.7%

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

      4. swap-sqr 99.6%

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

      5. unpow2 99.6%

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

   7. Simplified 99.6%

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

      1. unpow2 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. /-rgt-identity 99.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\left(r \cdot w\right) \cdot \color{blue}{\frac{r \cdot w}{1}}\right) \cdot -0.25 + -1.5\right) \]

       2. clear-num 99.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\left(r \cdot w\right) \cdot \color{blue}{\frac{1}{\frac{1}{r \cdot w}}}\right) \cdot -0.25 + -1.5\right) \]

       3. div-inv 99.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\frac{r \cdot w}{\frac{1}{r \cdot w}}} \cdot -0.25 + -1.5\right) \]

       4. associate-/r* 99.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(\frac{r \cdot w}{\color{blue}{\frac{\frac{1}{r}}{w}}} \cdot -0.25 + -1.5\right) \]

   11. Applied egg-rr 99.6%

       \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\frac{r \cdot w}{\frac{\frac{1}{r}}{w}}} \cdot -0.25 + -1.5\right) \]

   if -1e7 < v < 4.00000000000000022e-32

   1. Initial program 87.1%

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

   3. Simplified 87.1%

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

      1. add-sqr-sqrt 87.0%

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

      2. associate-/l* 87.0%

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

      3. sqrt-prod 39.4%

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

      4. sqrt-prod 39.5%

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

      5. sqrt-prod 20.1%

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

      6. add-sqr-sqrt 31.9%

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

      7. associate-*l* 31.9%

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

      8. add-sqr-sqrt 70.4%

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

      9. sqrt-prod 31.9%

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

      10. sqrt-prod 31.9%

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

      11. sqrt-prod 22.6%

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

      12. add-sqr-sqrt 42.0%

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

      13. associate-*l* 42.0%

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

      14. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{0.375} \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right) + -1.5\right) \]

   8. Taylor expanded in v around 0 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(-1 \cdot \left(r \cdot w\right)\right)}\right) + -1.5\right) \]
   9. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(\left(-1 \cdot r\right) \cdot w\right)}\right) + -1.5\right) \]

      2. neg-mul-1 99.8%

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

      3. *-commutative 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -10000000 \lor \neg \left(v \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + -0.25 \cdot \frac{r \cdot w}{\frac{\frac{1}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3100 \lor \neg \left(v \leq 4.8 \cdot 10^{-32}\right):\\ \;\;\;\;t\_1 + \left(-1.5 + -0.25 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 - 0.375 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -3100.0) (not (<= v 4.8e-32)))
     (+ t_1 (+ -1.5 (* -0.25 t_0)))
     (+ t_1 (- -1.5 (* 0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -3100.0) || !(v <= 4.8e-32)) {
		tmp = t_1 + (-1.5 + (-0.25 * t_0));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-3100.0d0)) .or. (.not. (v <= 4.8d-32))) then
        tmp = t_1 + ((-1.5d0) + ((-0.25d0) * t_0))
    else
        tmp = t_1 + ((-1.5d0) - (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -3100.0) || !(v <= 4.8e-32)) {
		tmp = t_1 + (-1.5 + (-0.25 * t_0));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -3100.0) or not (v <= 4.8e-32):
		tmp = t_1 + (-1.5 + (-0.25 * t_0))
	else:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -3100.0) || !(v <= 4.8e-32))
		tmp = Float64(t_1 + Float64(-1.5 + Float64(-0.25 * t_0)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -3100.0) || ~((v <= 4.8e-32)))
		tmp = t_1 + (-1.5 + (-0.25 * t_0));
	else
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -3100.0], N[Not[LessEqual[v, 4.8e-32]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 + N[(-0.25 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3100 or 4.8000000000000003e-32 < v

   1. Initial program 82.5%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in v around inf 84.7%

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

      1. *-commutative 84.7%

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

      2. unpow2 84.7%

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

      3. unpow2 84.7%

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

      4. swap-sqr 99.6%

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

      5. unpow2 99.6%

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

   7. Simplified 99.6%

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

      1. unpow2 99.6%

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

   9. Applied egg-rr 99.6%

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

   if -3100 < v < 4.8000000000000003e-32

   1. Initial program 87.1%

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

   3. Simplified 87.1%

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

      1. add-sqr-sqrt 87.0%

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

      2. associate-/l* 87.0%

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

      3. sqrt-prod 39.4%

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

      4. sqrt-prod 39.5%

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

      5. sqrt-prod 20.1%

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

      6. add-sqr-sqrt 31.9%

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

      7. associate-*l* 31.9%

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

      8. add-sqr-sqrt 70.4%

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

      9. sqrt-prod 31.9%

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

      10. sqrt-prod 31.9%

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

      11. sqrt-prod 22.6%

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

      12. add-sqr-sqrt 42.0%

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

      13. associate-*l* 42.0%

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

      14. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{0.375} \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right) + -1.5\right) \]

   8. Taylor expanded in v around 0 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(-1 \cdot \left(r \cdot w\right)\right)}\right) + -1.5\right) \]
   9. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(\left(-1 \cdot r\right) \cdot w\right)}\right) + -1.5\right) \]

      2. neg-mul-1 99.8%

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

      3. *-commutative 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3100 \lor \neg \left(v \leq 4.8 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (+ -1.5 (* (+ 0.375 (* v -0.25)) (* (* r w) (/ (* r w) (+ v -1.0)))))))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) * ((r * w) * ((r * w) / (v + -1.0)))));
}

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 + (v * (-0.25d0))) * ((r * w) * ((r * w) / (v + (-1.0d0))))))
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) * ((r * w) * ((r * w) / (v + -1.0)))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) * ((r * w) * ((r * w) / (v + -1.0)))))

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(v + -1.0))))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) * ((r * w) * ((r * w) / (v + -1.0)))));
end

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.6%

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

3. Simplified 88.9%

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

   1. add-sqr-sqrt 88.9%

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

   2. associate-/l* 88.9%

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

   3. sqrt-prod 42.9%

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

   4. sqrt-prod 42.9%

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

   5. sqrt-prod 23.0%

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

   6. add-sqr-sqrt 36.0%

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

   7. associate-*l* 36.0%

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

   8. add-sqr-sqrt 72.7%

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

   9. sqrt-prod 36.0%

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

   10. sqrt-prod 36.0%

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

   11. sqrt-prod 25.3%

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

   12. add-sqr-sqrt 46.4%

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

   13. associate-*l* 46.4%

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

   14. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in v around 0 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(0.375 + -0.25 \cdot v\right)} \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right) + -1.5\right) \]

8. Final simplification 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v + -1}\right)\right) \]
9. Add Preprocessing

Alternative 5: 93.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + (-0.25 * ((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) + ((-0.25d0) * ((r * w) * (r * w))))
end function

Java

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

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + (-0.25 * ((r * w) * (r * w))))

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + (-0.25 * ((r * w) * (r * w))));
end

Wolfram

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

Derivation

1. Initial program 84.6%

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

3. Simplified 88.9%

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

5. Taylor expanded in v around inf 77.3%

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

   1. *-commutative 77.3%

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

   2. unpow2 77.3%

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

   3. unpow2 77.3%

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

   4. swap-sqr 92.5%

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

   5. unpow2 92.5%

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

7. Simplified 92.5%

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

   1. unpow2 92.5%

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

9. Applied egg-rr 92.5%

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

10. Final simplification 92.5%

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

Reproduce

herbie shell --seed 2024170 
(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))