Rosa's TurbineBenchmark

Percentage Accurate: 85.1% → 99.8%

Time: 16.2s

Alternatives: 9

Speedup: 1.7×

• 53.5% 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: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 96.9%

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

   1. *-un-lft-identity 96.9%

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

   2. associate-*r* 99.7%

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

   3. times-frac 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. associate-/r* 98.3%

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

8. Applied egg-rr 98.3%

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

9. Taylor expanded in v around 0 91.9%

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

    1. +-commutative 91.9%

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

    2. mul-1-neg 91.9%

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

    3. sub-neg 91.9%

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

    4. div-sub 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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) + ((((v * (-0.25d0)) - (-0.375d0)) / (v + (-1.0d0))) / ((r * w) ** (-2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 96.9%

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

   1. *-un-lft-identity 96.9%

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

   2. div-inv 96.9%

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

   3. times-frac 92.5%

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

   4. associate-*r* 94.2%

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

   5. pow2 94.2%

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

   6. *-commutative 94.2%

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

6. Applied egg-rr 94.2%

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

   1. frac-2neg 94.2%

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

   2. metadata-eval 94.2%

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

   3. clear-num 94.2%

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

   4. frac-times 94.9%

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

   5. metadata-eval 94.9%

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

   6. pow-flip 94.9%

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

   7. metadata-eval 94.9%

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

8. Applied egg-rr 94.9%

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

   1. associate-*r/ 99.7%

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

   2. distribute-lft-neg-in 99.7%

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

   3. associate-/l* 99.7%

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

   4. neg-mul-1 99.7%

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

   5. distribute-lft-neg-in 99.7%

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

   6. associate-/r* 99.7%

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

   7. neg-sub0 99.7%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   8. fma-udef 99.7%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   9. *-commutative 99.7%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   10. +-commutative 99.7%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   11. associate--r+ 99.7%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. neg-sub0 99.7%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{-0.375 - v \cdot -0.25}{\color{blue}{0 - \left(1 - v\right)}}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   15. associate--r- 99.7%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{-0.375 - v \cdot -0.25}{\color{blue}{\left(0 - 1\right) + v}}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   16. metadata-eval 99.7%

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

   17. *-commutative 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 3: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -12500000 \lor \neg \left(v \leq 0.15\right):\\ \;\;\;\;t_0 + \left(-1.5 + \frac{r}{\frac{v}{w}} \cdot \frac{v \cdot -0.25 - -0.375}{\frac{1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (or (<= v -12500000.0) (not (<= v 0.15)))
     (+
      t_0
      (+ -1.5 (* (/ r (/ v w)) (/ (- (* v -0.25) -0.375) (/ 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 <= -12500000.0) || !(v <= 0.15)) {
		tmp = t_0 + (-1.5 + ((r / (v / w)) * (((v * -0.25) - -0.375) / (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 <= (-12500000.0d0)) .or. (.not. (v <= 0.15d0))) then
        tmp = t_0 + ((-1.5d0) + ((r / (v / w)) * (((v * (-0.25d0)) - (-0.375d0)) / (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 <= -12500000.0) || !(v <= 0.15)) {
		tmp = t_0 + (-1.5 + ((r / (v / w)) * (((v * -0.25) - -0.375) / (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 <= -12500000.0) or not (v <= 0.15):
		tmp = t_0 + (-1.5 + ((r / (v / w)) * (((v * -0.25) - -0.375) / (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(Float64(2.0 / r) / r)
	tmp = 0.0
	if ((v <= -12500000.0) || !(v <= 0.15))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r / Float64(v / w)) * Float64(Float64(Float64(v * -0.25) - -0.375) / Float64(1.0 / Float64(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 <= -12500000.0) || ~((v <= 0.15)))
		tmp = t_0 + (-1.5 + ((r / (v / w)) * (((v * -0.25) - -0.375) / (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[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[Or[LessEqual[v, -12500000.0], N[Not[LessEqual[v, 0.15]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(r / N[(v / w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(v * -0.25), $MachinePrecision] - -0.375), $MachinePrecision] / N[(1.0 / N[(r * w), $MachinePrecision]), $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 < -1.25e7 or 0.149999999999999994 < v

   1. Initial program 83.2%

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

   3. Simplified 96.6%

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

      1. *-un-lft-identity 96.6%

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

      2. div-inv 96.6%

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

      3. times-frac 89.0%

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

      4. associate-*r* 90.2%

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

      5. pow2 90.2%

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

      6. *-commutative 90.2%

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

   6. Applied egg-rr 90.2%

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

      1. frac-2neg 90.2%

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

      2. metadata-eval 90.2%

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

      3. clear-num 90.2%

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

      4. frac-times 91.4%

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

      5. metadata-eval 91.4%

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

      6. pow-flip 91.4%

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

      7. metadata-eval 91.4%

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

   8. Applied egg-rr 91.4%

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

      1. associate-*r/ 99.7%

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

      2. distribute-lft-neg-in 99.7%

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

      3. associate-/l* 99.7%

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

      4. neg-mul-1 99.7%

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

      5. distribute-lft-neg-in 99.7%

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

      6. associate-/r* 99.7%

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

      7. neg-sub0 99.7%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      8. fma-udef 99.7%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      9. *-commutative 99.7%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      10. +-commutative 99.7%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      11. associate--r+ 99.7%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. neg-sub0 99.7%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{-0.375 - v \cdot -0.25}{\color{blue}{0 - \left(1 - v\right)}}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      15. associate--r- 99.7%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{-0.375 - v \cdot -0.25}{\color{blue}{\left(0 - 1\right) + v}}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      16. metadata-eval 99.7%

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

      17. *-commutative 99.7%

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

   10. Simplified 99.7%

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

       1. div-inv 99.7%

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

       2. *-commutative 99.7%

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

       3. sqr-pow 99.7%

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

       4. times-frac 97.8%

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

       5. metadata-eval 97.8%

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

       6. inv-pow 97.8%

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

       7. *-commutative 97.8%

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

       8. +-commutative 97.8%

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

       9. metadata-eval 97.8%

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

       10. inv-pow 97.8%

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

       11. *-commutative 97.8%

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

   12. Applied egg-rr 97.8%

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

   13. Taylor expanded in v around inf 97.6%

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

       1. associate-/l* 95.1%

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

   15. Simplified 95.1%

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

   if -1.25e7 < v < 0.149999999999999994

   1. Initial program 85.6%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in v around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. *-commutative 75.7%

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

      3. unpow2 75.7%

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

      4. unpow2 75.7%

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

      5. swap-sqr 98.6%

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

      6. unpow2 98.6%

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

      7. *-commutative 98.6%

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

   7. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. unpow2 98.6%

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

   9. Applied egg-rr 98.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -12500000 \lor \neg \left(v \leq 0.15\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \frac{r}{\frac{v}{w}} \cdot \frac{v \cdot -0.25 - -0.375}{\frac{1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -12500000 \lor \neg \left(v \leq 0.15\right):\\ \;\;\;\;t_0 + \left(-1.5 + \frac{r \cdot w}{v} \cdot \frac{v \cdot -0.25 - -0.375}{\frac{1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (or (<= v -12500000.0) (not (<= v 0.15)))
     (+
      t_0
      (+ -1.5 (* (/ (* r w) v) (/ (- (* v -0.25) -0.375) (/ 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 <= -12500000.0) || !(v <= 0.15)) {
		tmp = t_0 + (-1.5 + (((r * w) / v) * (((v * -0.25) - -0.375) / (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 <= (-12500000.0d0)) .or. (.not. (v <= 0.15d0))) then
        tmp = t_0 + ((-1.5d0) + (((r * w) / v) * (((v * (-0.25d0)) - (-0.375d0)) / (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 <= -12500000.0) || !(v <= 0.15)) {
		tmp = t_0 + (-1.5 + (((r * w) / v) * (((v * -0.25) - -0.375) / (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 <= -12500000.0) or not (v <= 0.15):
		tmp = t_0 + (-1.5 + (((r * w) / v) * (((v * -0.25) - -0.375) / (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(Float64(2.0 / r) / r)
	tmp = 0.0
	if ((v <= -12500000.0) || !(v <= 0.15))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(Float64(r * w) / v) * Float64(Float64(Float64(v * -0.25) - -0.375) / Float64(1.0 / Float64(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 <= -12500000.0) || ~((v <= 0.15)))
		tmp = t_0 + (-1.5 + (((r * w) / v) * (((v * -0.25) - -0.375) / (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[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[Or[LessEqual[v, -12500000.0], N[Not[LessEqual[v, 0.15]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision] * N[(N[(N[(v * -0.25), $MachinePrecision] - -0.375), $MachinePrecision] / N[(1.0 / N[(r * w), $MachinePrecision]), $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 < -1.25e7 or 0.149999999999999994 < v

   1. Initial program 83.2%

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

   3. Simplified 96.6%

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

      1. *-un-lft-identity 96.6%

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

      2. div-inv 96.6%

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

      3. times-frac 89.0%

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

      4. associate-*r* 90.2%

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

      5. pow2 90.2%

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

      6. *-commutative 90.2%

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

   6. Applied egg-rr 90.2%

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

      1. frac-2neg 90.2%

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

      2. metadata-eval 90.2%

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

      3. clear-num 90.2%

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

      4. frac-times 91.4%

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

      5. metadata-eval 91.4%

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

      6. pow-flip 91.4%

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

      7. metadata-eval 91.4%

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

   8. Applied egg-rr 91.4%

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

      1. associate-*r/ 99.7%

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

      2. distribute-lft-neg-in 99.7%

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

      3. associate-/l* 99.7%

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

      4. neg-mul-1 99.7%

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

      5. distribute-lft-neg-in 99.7%

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

      6. associate-/r* 99.7%

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

      7. neg-sub0 99.7%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      8. fma-udef 99.7%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      9. *-commutative 99.7%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      10. +-commutative 99.7%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      11. associate--r+ 99.7%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. neg-sub0 99.7%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{-0.375 - v \cdot -0.25}{\color{blue}{0 - \left(1 - v\right)}}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      15. associate--r- 99.7%

          \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{-0.375 - v \cdot -0.25}{\color{blue}{\left(0 - 1\right) + v}}}{{\left(w \cdot r\right)}^{-2}}\right) \]

      16. metadata-eval 99.7%

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

      17. *-commutative 99.7%

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

   10. Simplified 99.7%

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

       1. div-inv 99.7%

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

       2. *-commutative 99.7%

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

       3. sqr-pow 99.7%

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

       4. times-frac 97.8%

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

       5. metadata-eval 97.8%

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

       6. inv-pow 97.8%

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

       7. *-commutative 97.8%

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

       8. +-commutative 97.8%

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

       9. metadata-eval 97.8%

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

       10. inv-pow 97.8%

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

       11. *-commutative 97.8%

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

   12. Applied egg-rr 97.8%

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

   13. Taylor expanded in v around inf 97.6%

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

   if -1.25e7 < v < 0.149999999999999994

   1. Initial program 85.6%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in v around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. *-commutative 75.7%

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

      3. unpow2 75.7%

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

      4. unpow2 75.7%

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

      5. swap-sqr 98.6%

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

      6. unpow2 98.6%

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

      7. *-commutative 98.6%

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

   7. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. unpow2 98.6%

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

   9. Applied egg-rr 98.6%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -12500000 \lor \neg \left(v \leq 0.15\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \frac{r \cdot w}{v} \cdot \frac{v \cdot -0.25 - -0.375}{\frac{1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + \frac{1}{r \cdot \frac{r}{2}}\\ \mathbf{elif}\;r \leq 2.7 \cdot 10^{+142}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot r\right) \cdot \frac{-0.375 + v \cdot 0.25}{\frac{1 - v}{w \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2e-39)
   (+ (- -1.5 (* 0.375 (* (* r w) (* r w)))) (/ 1.0 (* r (/ r 2.0))))
   (if (<= r 2.7e+142)
     (+
      -1.5
      (+
       (/ 2.0 (* r r))
       (* (* r r) (/ (+ -0.375 (* v 0.25)) (/ (- 1.0 v) (* w w))))))
     (+ (/ (/ 2.0 r) r) (- -1.5 (* 0.375 (* r (* w (* r w)))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-39) {
		tmp = (-1.5 - (0.375 * ((r * w) * (r * w)))) + (1.0 / (r * (r / 2.0)));
	} else if (r <= 2.7e+142) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * r) * ((-0.375 + (v * 0.25)) / ((1.0 - v) / (w * w)))));
	} else {
		tmp = ((2.0 / r) / r) + (-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) :: tmp
    if (r <= 2d-39) then
        tmp = ((-1.5d0) - (0.375d0 * ((r * w) * (r * w)))) + (1.0d0 / (r * (r / 2.0d0)))
    else if (r <= 2.7d+142) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((r * r) * (((-0.375d0) + (v * 0.25d0)) / ((1.0d0 - v) / (w * w)))))
    else
        tmp = ((2.0d0 / r) / r) + ((-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 tmp;
	if (r <= 2e-39) {
		tmp = (-1.5 - (0.375 * ((r * w) * (r * w)))) + (1.0 / (r * (r / 2.0)));
	} else if (r <= 2.7e+142) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * r) * ((-0.375 + (v * 0.25)) / ((1.0 - v) / (w * w)))));
	} else {
		tmp = ((2.0 / r) / r) + (-1.5 - (0.375 * (r * (w * (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2e-39:
		tmp = (-1.5 - (0.375 * ((r * w) * (r * w)))) + (1.0 / (r * (r / 2.0)))
	elif r <= 2.7e+142:
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * r) * ((-0.375 + (v * 0.25)) / ((1.0 - v) / (w * w)))))
	else:
		tmp = ((2.0 / r) / r) + (-1.5 - (0.375 * (r * (w * (r * w)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2e-39)
		tmp = Float64(Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))) + Float64(1.0 / Float64(r * Float64(r / 2.0))));
	elseif (r <= 2.7e+142)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(r * r) * Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(Float64(1.0 - v) / Float64(w * w))))));
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(w * Float64(r * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2e-39)
		tmp = (-1.5 - (0.375 * ((r * w) * (r * w)))) + (1.0 / (r * (r / 2.0)));
	elseif (r <= 2.7e+142)
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * r) * ((-0.375 + (v * 0.25)) / ((1.0 - v) / (w * w)))));
	else
		tmp = ((2.0 / r) / r) + (-1.5 - (0.375 * (r * (w * (r * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2e-39], N[(N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(r * N[(r / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 2.7e+142], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(r * r), $MachinePrecision] * N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(0.375 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.99999999999999986e-39

   1. Initial program 82.4%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in v around 0 75.5%

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

      1. *-commutative 75.5%

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

      2. *-commutative 75.5%

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

      3. unpow2 75.5%

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

      4. unpow2 75.5%

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

      5. swap-sqr 92.1%

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

      6. unpow2 92.1%

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

      7. *-commutative 92.1%

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

   7. Simplified 92.1%

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

      1. *-commutative 92.1%

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

      2. unpow2 92.1%

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

   9. Applied egg-rr 92.1%

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

       1. clear-num 92.0%

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

       2. inv-pow 92.0%

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

   11. Applied egg-rr 92.0%

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

       1. unpow-1 92.0%

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

       2. associate-/r/ 92.1%

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

   13. Simplified 92.1%

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

   if 1.99999999999999986e-39 < r < 2.69999999999999983e142

   1. Initial program 92.6%

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

   3. Simplified 99.7%

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

   if 2.69999999999999983e142 < r

   1. Initial program 84.5%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 60.6%

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

      1. *-commutative 60.6%

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

      2. *-commutative 60.6%

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

      3. unpow2 60.6%

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

      4. unpow2 60.6%

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

      5. swap-sqr 84.1%

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

      6. unpow2 84.1%

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

      7. *-commutative 84.1%

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

   7. Simplified 84.1%

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

      1. *-commutative 84.1%

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

      2. metadata-eval 84.1%

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

      3. metadata-eval 84.1%

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

      4. pow-flip 84.2%

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

      5. metadata-eval 84.2%

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

   9. Applied egg-rr 84.2%

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

       1. pow-flip 84.1%

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

       2. metadata-eval 84.1%

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

       3. pow2 84.1%

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

       4. associate-*r* 84.2%

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

       5. *-commutative 84.2%

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

   11. Applied egg-rr 84.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + \frac{1}{r \cdot \frac{r}{2}}\\ \mathbf{elif}\;r \leq 2.7 \cdot 10^{+142}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot r\right) \cdot \frac{-0.375 + v \cdot 0.25}{\frac{1 - v}{w \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 - ((r * w) * (((-0.375 - (v * -0.25)) / (v + -1.0)) / (1.0 / (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) - ((r * w) * ((((-0.375d0) - (v * (-0.25d0))) / (v + (-1.0d0))) / (1.0d0 / (r * w)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 96.9%

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

   1. *-un-lft-identity 96.9%

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

   2. div-inv 96.9%

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

   3. times-frac 92.5%

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

   4. associate-*r* 94.2%

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

   5. pow2 94.2%

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

   6. *-commutative 94.2%

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

6. Applied egg-rr 94.2%

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

   1. frac-2neg 94.2%

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

   2. metadata-eval 94.2%

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

   3. clear-num 94.2%

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

   4. frac-times 94.9%

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

   5. metadata-eval 94.9%

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

   6. pow-flip 94.9%

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

   7. metadata-eval 94.9%

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

8. Applied egg-rr 94.9%

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

   1. associate-*r/ 99.7%

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

   2. distribute-lft-neg-in 99.7%

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

   3. associate-/l* 99.7%

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

   4. neg-mul-1 99.7%

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

   5. distribute-lft-neg-in 99.7%

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

   6. associate-/r* 99.7%

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

   7. neg-sub0 99.7%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{\color{blue}{0 - \mathsf{fma}\left(v, -0.25, 0.375\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   8. fma-udef 99.7%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \color{blue}{\left(v \cdot -0.25 + 0.375\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   9. *-commutative 99.7%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \left(\color{blue}{-0.25 \cdot v} + 0.375\right)}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   10. +-commutative 99.7%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{0 - \color{blue}{\left(0.375 + -0.25 \cdot v\right)}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   11. associate--r+ 99.7%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{\color{blue}{\left(0 - 0.375\right) - -0.25 \cdot v}}{-\left(1 - v\right)}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. neg-sub0 99.7%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{-0.375 - v \cdot -0.25}{\color{blue}{0 - \left(1 - v\right)}}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   15. associate--r- 99.7%

       \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\frac{-0.375 - v \cdot -0.25}{\color{blue}{\left(0 - 1\right) + v}}}{{\left(w \cdot r\right)}^{-2}}\right) \]

   16. metadata-eval 99.7%

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

   17. *-commutative 99.7%

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

10. Simplified 99.7%

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

    1. *-un-lft-identity 99.7%

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

    2. *-commutative 99.7%

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

    3. sqr-pow 99.7%

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

    4. times-frac 99.7%

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

    5. metadata-eval 99.7%

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

    6. sqrt-pow1 75.8%

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

    7. sqrt-div 75.8%

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

    8. pow-flip 75.8%

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

    9. metadata-eval 75.8%

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

    10. pow2 75.8%

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

    11. sqrt-prod 55.2%

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

    12. add-sqr-sqrt 99.7%

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

    13. *-commutative 99.7%

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

    14. +-commutative 99.7%

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

    15. metadata-eval 99.7%

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

    16. inv-pow 99.7%

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

    17. *-commutative 99.7%

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

12. Applied egg-rr 99.7%

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

13. Final simplification 99.7%

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

Alternative 7: 93.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((-1.5d0) - (0.375d0 * ((r * w) * (r * w)))) + (1.0d0 / (r * (r / 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 96.9%

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

5. Taylor expanded in v around 0 74.2%

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

   1. *-commutative 74.2%

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

   2. *-commutative 74.2%

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

   3. unpow2 74.2%

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

   4. unpow2 74.2%

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

   5. swap-sqr 89.4%

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

   6. unpow2 89.4%

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

   7. *-commutative 89.4%

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

7. Simplified 89.4%

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

   1. *-commutative 89.4%

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

   2. unpow2 89.4%

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

9. Applied egg-rr 89.4%

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

    1. clear-num 89.4%

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

    2. inv-pow 89.4%

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

11. Applied egg-rr 89.4%

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

    1. unpow-1 89.4%

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

    2. associate-/r/ 89.4%

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

13. Simplified 89.4%

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

14. Final simplification 89.4%

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

Alternative 8: 92.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / r) / r) + ((-1.5d0) - (0.375d0 * (w * (r * (r * w)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 96.9%

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

5. Taylor expanded in v around 0 74.2%

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

   1. *-commutative 74.2%

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

   2. *-commutative 74.2%

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

   3. unpow2 74.2%

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

   4. unpow2 74.2%

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

   5. swap-sqr 89.4%

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

   6. unpow2 89.4%

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

   7. *-commutative 89.4%

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

7. Simplified 89.4%

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

   1. *-commutative 89.4%

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

   2. metadata-eval 89.4%

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

   3. metadata-eval 89.4%

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

   4. pow-flip 89.4%

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

   5. metadata-eval 89.4%

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

9. Applied egg-rr 89.4%

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

    1. pow-flip 89.4%

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

    2. metadata-eval 89.4%

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

    3. pow2 89.4%

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

    4. associate-*l* 87.3%

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

    5. *-commutative 87.3%

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

11. Applied egg-rr 87.3%

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

12. Final simplification 87.3%

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

Alternative 9: 93.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / r) / r) + ((-1.5d0) - (0.375d0 * ((r * w) * (r * w))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Simplified 96.9%

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

5. Taylor expanded in v around 0 74.2%

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

   1. *-commutative 74.2%

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

   2. *-commutative 74.2%

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

   3. unpow2 74.2%

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

   4. unpow2 74.2%

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

   5. swap-sqr 89.4%

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

   6. unpow2 89.4%

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

   7. *-commutative 89.4%

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

7. Simplified 89.4%

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

   1. *-commutative 89.4%

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

   2. unpow2 89.4%

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

9. Applied egg-rr 89.4%

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

10. Final simplification 89.4%

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

Reproduce

herbie shell --seed 2024019 
(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))