Rosa's TurbineBenchmark

Percentage Accurate: 85.2% → 99.7%

Time: 8.7s

Alternatives: 9

Speedup: 1.5×

• 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: 85.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. associate--l- 87.8%

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

   2. +-commutative 87.8%

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

   3. associate--l+ 87.8%

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

   4. +-commutative 87.8%

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

   5. associate--r+ 87.8%

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

   6. metadata-eval 87.8%

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

   7. associate-*l/ 89.6%

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

   8. *-commutative 89.6%

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

   9. *-commutative 89.6%

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

   10. *-commutative 89.6%

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

3. Simplified 89.6%

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

   1. add-sqr-sqrt 89.6%

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

   2. pow2 89.6%

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

   3. *-commutative 89.6%

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

   4. sqrt-prod 89.5%

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

   5. +-commutative 89.5%

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

   6. fma-def 89.5%

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

   7. associate-*r* 85.9%

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

   8. unswap-sqr 99.4%

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

   9. sqrt-prod 48.7%

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

   10. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt{\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}} \cdot \left(r \cdot w\right)\right)}^{2}}\right) \]
6. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - {\left(\left(r \cdot w\right) \cdot \sqrt{\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}}\right)}^{2}\right) \]

Alternative 2: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 3.85 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(t_0 + 3\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 3.85e-31)
     (+ (- (+ t_0 3.0) (* 0.375 (* (* r w) (* r w)))) -4.5)
     (+
      t_0
      (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* r (* w w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3.85e-31) {
		tmp = ((t_0 + 3.0) - (0.375 * ((r * w) * (r * w)))) + -4.5;
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * 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 (r <= 3.85d-31) then
        tmp = ((t_0 + 3.0d0) - (0.375d0 * ((r * w) * (r * w)))) + (-4.5d0)
    else
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (r * (w * 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 (r <= 3.85e-31) {
		tmp = ((t_0 + 3.0) - (0.375 * ((r * w) * (r * w)))) + -4.5;
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 3.85e-31:
		tmp = ((t_0 + 3.0) - (0.375 * ((r * w) * (r * w)))) + -4.5
	else:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 3.85e-31)
		tmp = Float64(Float64(Float64(t_0 + 3.0) - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))) + -4.5);
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 3.85e-31)
		tmp = ((t_0 + 3.0) - (0.375 * ((r * w) * (r * w)))) + -4.5;
	else
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	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, 3.85e-31], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.85000000000000006e-31

   1. Initial program 86.9%

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

      1. sub-neg 86.9%

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

      2. associate-/l* 88.9%

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

      3. cancel-sign-sub-inv 88.9%

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

      4. metadata-eval 88.9%

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

      5. *-commutative 88.9%

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

      6. *-commutative 88.9%

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

      7. metadata-eval 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in v around 0 84.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 84.1%

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

      2. unpow2 84.1%

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

      3. unpow2 84.1%

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

      4. swap-sqr 95.5%

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

      5. unpow2 95.5%

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

      6. *-commutative 95.5%

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

   6. Simplified 95.5%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.375 \cdot {\left(w \cdot r\right)}^{2}}\right) + -4.5 \]
   7. Step-by-step derivation

      1. unpow2 95.5%

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

   8. Applied egg-rr 95.5%

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

   if 3.85000000000000006e-31 < r

   1. Initial program 90.3%

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

      1. associate--l- 90.3%

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

      2. +-commutative 90.3%

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

      3. associate--l+ 90.3%

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

      4. +-commutative 90.3%

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

      5. associate--r+ 90.3%

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

      6. metadata-eval 90.3%

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

      7. associate-*l/ 91.8%

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

      8. *-commutative 91.8%

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

      9. *-commutative 91.8%

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

      10. *-commutative 91.8%

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

   3. Simplified 91.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.85 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 96.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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 + (v * (-0.25d0))) / (1.0d0 - v))))
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 + (v * -0.25)) / (1.0 - v))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. associate--l- 87.8%

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

   2. +-commutative 87.8%

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

   3. associate--l+ 87.8%

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

   4. +-commutative 87.8%

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

   5. associate--r+ 87.8%

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

   6. metadata-eval 87.8%

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

   7. associate-*l/ 89.6%

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

   8. *-commutative 89.6%

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

   9. *-commutative 89.6%

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

   10. *-commutative 89.6%

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

3. Simplified 89.6%

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

4. Taylor expanded in r around 0 89.6%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left({w}^{2} \cdot r\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
5. Step-by-step derivation

   1. unpow2 89.6%

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

   2. associate-*l* 98.3%

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

6. Simplified 98.3%

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

7. Final simplification 98.3%

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

Alternative 4: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.78 \cdot 10^{-99}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375 - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.78e-99)
   (* 2.0 (/ (/ 1.0 r) r))
   (+ (/ 2.0 (* r r)) (- (* (* (* r r) (* w w)) -0.375) 1.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.78e-99) {
		tmp = 2.0 * ((1.0 / r) / r);
	} else {
		tmp = (2.0 / (r * r)) + ((((r * r) * (w * w)) * -0.375) - 1.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.78d-99) then
        tmp = 2.0d0 * ((1.0d0 / r) / r)
    else
        tmp = (2.0d0 / (r * r)) + ((((r * r) * (w * w)) * (-0.375d0)) - 1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.78e-99) {
		tmp = 2.0 * ((1.0 / r) / r);
	} else {
		tmp = (2.0 / (r * r)) + ((((r * r) * (w * w)) * -0.375) - 1.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.78e-99:
		tmp = 2.0 * ((1.0 / r) / r)
	else:
		tmp = (2.0 / (r * r)) + ((((r * r) * (w * w)) * -0.375) - 1.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.78e-99)
		tmp = Float64(2.0 * Float64(Float64(1.0 / r) / r));
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(Float64(r * r) * Float64(w * w)) * -0.375) - 1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.78e-99)
		tmp = 2.0 * ((1.0 / r) / r);
	else
		tmp = (2.0 / (r * r)) + ((((r * r) * (w * w)) * -0.375) - 1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.78e-99], N[(2.0 * N[(N[(1.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.7800000000000001e-99

   1. Initial program 86.5%

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

      1. sub-neg 86.5%

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

      2. associate-/l* 88.6%

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

      3. cancel-sign-sub-inv 88.6%

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

      4. metadata-eval 88.6%

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

      5. *-commutative 88.6%

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

      6. *-commutative 88.6%

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

      7. metadata-eval 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in v around 0 83.5%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 83.5%

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

      2. unpow2 83.5%

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

      3. unpow2 83.5%

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

      4. swap-sqr 95.7%

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

      5. unpow2 95.7%

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

      6. *-commutative 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in r around 0 61.8%

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

      1. unpow2 61.8%

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

   9. Simplified 61.8%

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

       1. div-inv 61.8%

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

       2. pow2 61.8%

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

       3. pow-flip 61.9%

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

       4. metadata-eval 61.9%

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

   11. Applied egg-rr 61.9%

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

       1. metadata-eval 61.9%

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

       2. pow-flip 61.8%

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

       3. pow2 61.8%

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

       4. associate-/r* 61.8%

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

   13. Applied egg-rr 61.8%

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

   if 1.7800000000000001e-99 < r

   1. Initial program 90.6%

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

      1. sub-neg 90.6%

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

      2. +-commutative 90.6%

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

      3. associate--l+ 90.6%

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

      4. associate-/l* 91.8%

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

      5. distribute-neg-frac 91.8%

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

      6. associate-/r/ 91.8%

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

      7. fma-def 91.8%

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

      8. sub-neg 91.8%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in v around 0 83.3%

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

      1. associate--l+ 83.3%

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

      2. associate-*r/ 83.3%

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

      3. metadata-eval 83.3%

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

      4. unpow2 83.3%

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

      5. *-commutative 83.3%

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

      6. unpow2 83.3%

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

      7. unpow2 83.3%

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

   6. Simplified 83.3%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.78 \cdot 10^{-99}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375 - 1.5\right)\\ \end{array} \]

Alternative 5: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{2}{r \cdot r} + 3\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \end{array} \]

FPCore

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

C

double code(double v, double w, double r) {
	return (((2.0 / (r * r)) + 3.0) - (0.375 * ((r * w) * (r * w)))) + -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 = (((2.0d0 / (r * r)) + 3.0d0) - (0.375d0 * ((r * w) * (r * w)))) + (-4.5d0)
end function

Java

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

Python

def code(v, w, r):
	return (((2.0 / (r * r)) + 3.0) - (0.375 * ((r * w) * (r * w)))) + -4.5

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (((2.0 / (r * r)) + 3.0) - (0.375 * ((r * w) * (r * w)))) + -4.5;
end

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. sub-neg 87.8%

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

   2. associate-/l* 89.6%

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

   3. cancel-sign-sub-inv 89.6%

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

   4. metadata-eval 89.6%

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

   5. *-commutative 89.6%

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

   6. *-commutative 89.6%

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

   7. metadata-eval 89.6%

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

3. Simplified 89.6%

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

4. Taylor expanded in v around 0 83.5%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)}\right) + -4.5 \]
5. Step-by-step derivation

   1. *-commutative 83.5%

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

   2. unpow2 83.5%

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

   3. unpow2 83.5%

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

   4. swap-sqr 94.7%

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

   5. unpow2 94.7%

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

   6. *-commutative 94.7%

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

6. Simplified 94.7%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.375 \cdot {\left(w \cdot r\right)}^{2}}\right) + -4.5 \]
7. Step-by-step derivation

   1. unpow2 94.7%

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

8. Applied egg-rr 94.7%

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

9. Final simplification 94.7%

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

Alternative 6: 68.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq 5.4 \cdot 10^{-103}:\\ \;\;\;\;t_0 + -1.5\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= w 5.4e-103) (+ t_0 -1.5) (+ t_0 (* (* r r) (* w (* w -0.375)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (w <= 5.4e-103) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + ((r * r) * (w * (w * -0.375)));
	}
	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 (w <= 5.4d-103) then
        tmp = t_0 + (-1.5d0)
    else
        tmp = t_0 + ((r * r) * (w * (w * (-0.375d0))))
    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 (w <= 5.4e-103) {
		tmp = t_0 + -1.5;
	} else {
		tmp = t_0 + ((r * r) * (w * (w * -0.375)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if w <= 5.4e-103:
		tmp = t_0 + -1.5
	else:
		tmp = t_0 + ((r * r) * (w * (w * -0.375)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (w <= 5.4e-103)
		tmp = Float64(t_0 + -1.5);
	else
		tmp = Float64(t_0 + Float64(Float64(r * r) * Float64(w * Float64(w * -0.375))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (w <= 5.4e-103)
		tmp = t_0 + -1.5;
	else
		tmp = t_0 + ((r * r) * (w * (w * -0.375)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 5.4e-103], N[(t$95$0 + -1.5), $MachinePrecision], N[(t$95$0 + N[(N[(r * r), $MachinePrecision] * N[(w * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 5.40000000000000019e-103

   1. Initial program 90.5%

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

      1. sub-neg 90.5%

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

      2. +-commutative 90.5%

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

      3. associate--l+ 90.5%

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

      4. associate-/l* 91.1%

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

      5. distribute-neg-frac 91.1%

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

      6. associate-/r/ 91.1%

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

      7. fma-def 91.1%

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

      8. sub-neg 91.1%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in r around 0 67.8%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   5. Step-by-step derivation

      1. sub-neg 67.8%

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

      2. associate-*r/ 67.8%

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

      3. metadata-eval 67.8%

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

      4. unpow2 67.8%

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

      5. metadata-eval 67.8%

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

   6. Simplified 67.8%

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

   if 5.40000000000000019e-103 < w

   1. Initial program 81.9%

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

      1. sub-neg 81.9%

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

      2. +-commutative 81.9%

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

      3. associate--l+ 81.9%

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

      4. associate-/l* 86.5%

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

      5. distribute-neg-frac 86.5%

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

      6. associate-/r/ 86.5%

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

      7. fma-def 86.6%

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

      8. sub-neg 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in v around 0 79.7%

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

      1. associate--l+ 79.7%

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

      2. associate-*r/ 79.7%

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

      3. metadata-eval 79.7%

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

      4. unpow2 79.7%

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

      5. *-commutative 79.7%

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

      6. fma-neg 79.7%

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

      7. *-commutative 79.7%

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

      8. unpow2 79.7%

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

      9. unpow2 79.7%

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

      10. swap-sqr 92.0%

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

      11. unpow2 92.0%

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

      12. *-commutative 92.0%

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

      13. metadata-eval 92.0%

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

   6. Simplified 92.0%

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

      1. unpow2 92.0%

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

   8. Applied egg-rr 92.0%

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

   9. Taylor expanded in w around inf 79.6%

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

       1. unpow2 79.6%

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

       2. unpow2 79.6%

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

       3. associate-*r* 79.6%

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

       4. associate-*r* 79.6%

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

       5. *-commutative 79.6%

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

   11. Simplified 79.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 5.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \]

Alternative 7: 43.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\frac{1}{r}}{r} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (* 2.0 (/ (/ 1.0 r) r)))

C

double code(double v, double w, double r) {
	return 2.0 * ((1.0 / r) / r);
}

Fortran

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

Java

public static double code(double v, double w, double r) {
	return 2.0 * ((1.0 / r) / r);
}

Python

def code(v, w, r):
	return 2.0 * ((1.0 / r) / r)

Julia

function code(v, w, r)
	return Float64(2.0 * Float64(Float64(1.0 / r) / r))
end

MATLAB

function tmp = code(v, w, r)
	tmp = 2.0 * ((1.0 / r) / r);
end

Wolfram

code[v_, w_, r_] := N[(2.0 * N[(N[(1.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.8%

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

   1. sub-neg 87.8%

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

   2. associate-/l* 89.6%

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

   3. cancel-sign-sub-inv 89.6%

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

   4. metadata-eval 89.6%

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

   5. *-commutative 89.6%

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

   6. *-commutative 89.6%

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

   7. metadata-eval 89.6%

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

3. Simplified 89.6%

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

4. Taylor expanded in v around 0 83.5%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)}\right) + -4.5 \]
5. Step-by-step derivation

   1. *-commutative 83.5%

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

   2. unpow2 83.5%

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

   3. unpow2 83.5%

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

   4. swap-sqr 94.7%

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

   5. unpow2 94.7%

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

   6. *-commutative 94.7%

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

6. Simplified 94.7%

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

7. Taylor expanded in r around 0 48.5%

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

   1. unpow2 48.5%

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

9. Simplified 48.5%

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

    1. div-inv 48.5%

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

    2. pow2 48.5%

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

    3. pow-flip 48.6%

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

    4. metadata-eval 48.6%

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

11. Applied egg-rr 48.6%

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

    1. metadata-eval 48.6%

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

    2. pow-flip 48.5%

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

    3. pow2 48.5%

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

    4. associate-/r* 48.5%

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

13. Applied egg-rr 48.5%

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

14. Final simplification 48.5%

    \[\leadsto 2 \cdot \frac{\frac{1}{r}}{r} \]

Alternative 8: 56.9% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. sub-neg 87.8%

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

   2. +-commutative 87.8%

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

   3. associate--l+ 87.8%

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

   4. associate-/l* 89.6%

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

   5. distribute-neg-frac 89.6%

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

   6. associate-/r/ 89.6%

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

   7. fma-def 89.6%

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

   8. sub-neg 89.6%

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

3. Simplified 86.0%

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

4. Taylor expanded in r around 0 61.5%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
5. Step-by-step derivation

   1. sub-neg 61.5%

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

   2. associate-*r/ 61.5%

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

   3. metadata-eval 61.5%

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

   4. unpow2 61.5%

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

   5. metadata-eval 61.5%

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

6. Simplified 61.5%

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

7. Final simplification 61.5%

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

Alternative 9: 43.4% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. sub-neg 87.8%

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

   2. associate-/l* 89.6%

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

   3. cancel-sign-sub-inv 89.6%

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

   4. metadata-eval 89.6%

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

   5. *-commutative 89.6%

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

   6. *-commutative 89.6%

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

   7. metadata-eval 89.6%

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

3. Simplified 89.6%

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

4. Taylor expanded in v around 0 83.5%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)}\right) + -4.5 \]
5. Step-by-step derivation

   1. *-commutative 83.5%

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

   2. unpow2 83.5%

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

   3. unpow2 83.5%

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

   4. swap-sqr 94.7%

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

   5. unpow2 94.7%

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

   6. *-commutative 94.7%

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

6. Simplified 94.7%

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

7. Taylor expanded in r around 0 48.5%

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

   1. unpow2 48.5%

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

9. Simplified 48.5%

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

10. Final simplification 48.5%

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

Reproduce

herbie shell --seed 2023252 
(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))