Rosa's TurbineBenchmark

Percentage Accurate: 85.3% → 99.8%

Time: 15.5s

Alternatives: 5

Speedup: 1.2×

• 52.6% 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: 85.3% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 96.4%

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

   1. frac-2neg 96.4%

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

   2. *-commutative 96.4%

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

   3. associate-*r* 87.9%

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

   4. div-inv 87.8%

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

   5. associate-*r* 96.4%

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

   6. *-commutative 96.4%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

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

   4. fma-udef 99.8%

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

   5. *-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate--r+ 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

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

   12. associate--r- 99.8%

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

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. unpow2 99.8%

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

   3. times-frac 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. *-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-/r* 99.8%

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

    2. div-inv 99.8%

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

12. Applied egg-rr 99.8%

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

    1. un-div-inv 99.8%

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

14. Applied egg-rr 99.8%

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

15. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := v \cdot -0.25 - -0.375\\ t_1 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -720000000 \lor \neg \left(v \leq 0.1\right):\\ \;\;\;\;t\_1 + \left(-1.5 + \frac{t\_0}{\frac{\frac{v}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 + \frac{t\_0}{\frac{\frac{-1}{r \cdot w}}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- (* v -0.25) -0.375)) (t_1 (/ (/ 2.0 r) r)))
   (if (or (<= v -720000000.0) (not (<= v 0.1)))
     (+ t_1 (+ -1.5 (/ t_0 (/ (/ v (* r w)) (* r w)))))
     (+ t_1 (+ -1.5 (/ t_0 (/ (/ -1.0 (* r w)) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = (v * -0.25) - -0.375;
	double t_1 = (2.0 / r) / r;
	double tmp;
	if ((v <= -720000000.0) || !(v <= 0.1)) {
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))));
	} else {
		tmp = t_1 + (-1.5 + (t_0 / ((-1.0 / (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) :: t_1
    real(8) :: tmp
    t_0 = (v * (-0.25d0)) - (-0.375d0)
    t_1 = (2.0d0 / r) / r
    if ((v <= (-720000000.0d0)) .or. (.not. (v <= 0.1d0))) then
        tmp = t_1 + ((-1.5d0) + (t_0 / ((v / (r * w)) / (r * w))))
    else
        tmp = t_1 + ((-1.5d0) + (t_0 / (((-1.0d0) / (r * w)) / (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (v * -0.25) - -0.375;
	double t_1 = (2.0 / r) / r;
	double tmp;
	if ((v <= -720000000.0) || !(v <= 0.1)) {
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))));
	} else {
		tmp = t_1 + (-1.5 + (t_0 / ((-1.0 / (r * w)) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (v * -0.25) - -0.375
	t_1 = (2.0 / r) / r
	tmp = 0
	if (v <= -720000000.0) or not (v <= 0.1):
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))))
	else:
		tmp = t_1 + (-1.5 + (t_0 / ((-1.0 / (r * w)) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(v * -0.25) - -0.375)
	t_1 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if ((v <= -720000000.0) || !(v <= 0.1))
		tmp = Float64(t_1 + Float64(-1.5 + Float64(t_0 / Float64(Float64(v / Float64(r * w)) / Float64(r * w)))));
	else
		tmp = Float64(t_1 + Float64(-1.5 + Float64(t_0 / Float64(Float64(-1.0 / Float64(r * w)) / Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (v * -0.25) - -0.375;
	t_1 = (2.0 / r) / r;
	tmp = 0.0;
	if ((v <= -720000000.0) || ~((v <= 0.1)))
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))));
	else
		tmp = t_1 + (-1.5 + (t_0 / ((-1.0 / (r * w)) / (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(v * -0.25), $MachinePrecision] - -0.375), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[Or[LessEqual[v, -720000000.0], N[Not[LessEqual[v, 0.1]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 + N[(t$95$0 / N[(N[(v / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 + N[(t$95$0 / N[(N[(-1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -7.2e8 or 0.10000000000000001 < v

   1. Initial program 79.1%

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

   3. Simplified 97.4%

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

      1. frac-2neg 97.4%

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

      2. *-commutative 97.4%

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

      3. associate-*r* 88.0%

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

      4. div-inv 88.0%

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

      5. associate-*r* 97.3%

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

      6. *-commutative 97.3%

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

      7. associate-*r* 99.8%

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

      8. pow2 99.8%

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

      9. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. neg-sub0 99.8%

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

      4. fma-udef 99.8%

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

      5. *-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate--r+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-neg-frac 99.8%

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

      11. neg-sub0 99.8%

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

      12. associate--r- 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. *-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-un-lft-identity 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in v around inf 99.5%

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

   if -7.2e8 < v < 0.10000000000000001

   1. Initial program 87.7%

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

   3. Simplified 95.7%

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

      1. frac-2neg 95.7%

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

      2. *-commutative 95.7%

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

      3. associate-*r* 87.7%

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

      4. div-inv 87.7%

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

      5. associate-*r* 95.7%

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

      6. *-commutative 95.7%

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

      7. associate-*r* 99.8%

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

      8. pow2 99.8%

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

      9. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. neg-sub0 99.8%

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

      4. fma-udef 99.8%

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

      5. *-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate--r+ 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-neg-frac 99.8%

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

      11. neg-sub0 99.8%

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

      12. associate--r- 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. *-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-un-lft-identity 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in v around 0 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -720000000 \lor \neg \left(v \leq 0.1\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \frac{v \cdot -0.25 - -0.375}{\frac{\frac{v}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \frac{v \cdot -0.25 - -0.375}{\frac{\frac{-1}{r \cdot w}}{r \cdot w}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 96.4%

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

   1. frac-2neg 96.4%

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

   2. *-commutative 96.4%

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

   3. associate-*r* 87.9%

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

   4. div-inv 87.8%

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

   5. associate-*r* 96.4%

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

   6. *-commutative 96.4%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

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

   4. fma-udef 99.8%

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

   5. *-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate--r+ 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

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

   12. associate--r- 99.8%

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

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. unpow2 99.8%

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

   3. times-frac 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. *-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

function tmp = code(v, w, r)
	tmp = ((2.0 / r) / r) + (-1.5 + ((r * w) * (((v * -0.25) - -0.375) / ((v + -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[(v * -0.25), $MachinePrecision] - -0.375), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.9%

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

3. Simplified 96.4%

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

   1. frac-2neg 96.4%

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

   2. *-commutative 96.4%

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

   3. associate-*r* 87.9%

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

   4. div-inv 87.8%

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

   5. associate-*r* 96.4%

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

   6. *-commutative 96.4%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

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

   4. fma-udef 99.8%

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

   5. *-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate--r+ 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

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

   12. associate--r- 99.8%

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

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. unpow2 99.8%

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

   3. times-frac 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. *-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-/r* 99.8%

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

    2. div-inv 99.8%

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

12. Applied egg-rr 99.8%

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

    1. *-commutative 99.8%

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

    2. frac-times 99.8%

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

    3. metadata-eval 99.8%

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

    4. div-inv 99.8%

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

    5. associate-/r/ 99.8%

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

14. Applied egg-rr 99.8%

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

15. Final simplification 99.8%

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

Alternative 5: 81.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 96.4%

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

   1. frac-2neg 96.4%

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

   2. *-commutative 96.4%

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

   3. associate-*r* 87.9%

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

   4. div-inv 87.8%

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

   5. associate-*r* 96.4%

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

   6. *-commutative 96.4%

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

   7. associate-*r* 99.8%

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

   8. pow2 99.8%

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

   9. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. neg-sub0 99.8%

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

   4. fma-udef 99.8%

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

   5. *-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate--r+ 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. distribute-neg-frac 99.8%

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

   11. neg-sub0 99.8%

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

   12. associate--r- 99.8%

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

   13. metadata-eval 99.8%

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

   14. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. unpow2 99.8%

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

   3. times-frac 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. *-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.8%

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

    2. *-un-lft-identity 99.8%

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

12. Applied egg-rr 99.8%

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

13. Taylor expanded in v around 0 80.1%

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

14. Final simplification 80.1%

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

Reproduce

herbie shell --seed 2024033 
(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))