Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.8%

Time: 14.1s

Alternatives: 12

Speedup: 1.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Simplified 96.0%

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

   1. associate-*r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 - (fma(v, -0.25, 0.375) * ((r * w) / ((1.0 - v) / (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(r * w) / Float64(Float64(1.0 - v) / 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[(r * w), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

3. Simplified 96.0%

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

   1. associate-*r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. clear-num 99.8%

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

   3. un-div-inv 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Simplified 87.5%

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

4. Taylor expanded in r around 0 78.7%

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

   1. unpow2 78.7%

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

   2. unpow2 78.7%

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

   3. swap-sqr 99.7%

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

   4. unpow2 99.7%

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

6. Simplified 99.7%

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

   1. unpow2 93.8%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -1.05:\\ \;\;\;\;t_0 + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 7.2 \cdot 10^{-19}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{r \cdot \left(w \cdot -0.375\right)}{\frac{\frac{1}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (<= v -1.05)
     (+ t_0 (- -1.5 (* 0.25 (* (* r w) (* r w)))))
     (if (<= v 7.2e-19)
       (+ -1.5 (+ (/ 2.0 (* r r)) (/ (* r (* w -0.375)) (/ (/ 1.0 r) w))))
       (+ t_0 (- -1.5 (* 0.25 (* w (* r (* r w))))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if (v <= -1.05) {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	} else if (v <= 7.2e-19) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * -0.375)) / ((1.0 / r) / w)));
	} else {
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (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 <= (-1.05d0)) then
        tmp = t_0 + ((-1.5d0) - (0.25d0 * ((r * w) * (r * w))))
    else if (v <= 7.2d-19) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((r * (w * (-0.375d0))) / ((1.0d0 / r) / w)))
    else
        tmp = t_0 + ((-1.5d0) - (0.25d0 * (w * (r * (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 <= -1.05) {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	} else if (v <= 7.2e-19) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * -0.375)) / ((1.0 / r) / w)));
	} else {
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	tmp = 0
	if v <= -1.05:
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))))
	elif v <= 7.2e-19:
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * -0.375)) / ((1.0 / r) / w)))
	else:
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (r * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if (v <= -1.05)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * Float64(Float64(r * w) * Float64(r * w)))));
	elseif (v <= 7.2e-19)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(r * Float64(w * -0.375)) / Float64(Float64(1.0 / r) / w))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * Float64(w * Float64(r * 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 <= -1.05)
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	elseif (v <= 7.2e-19)
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * -0.375)) / ((1.0 / r) / w)));
	else
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (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[LessEqual[v, -1.05], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 7.2e-19], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.05000000000000004

   1. Initial program 72.7%

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

   3. Simplified 97.0%

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

      1. associate-*r* 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. times-frac 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in v around inf 75.6%

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

      1. unpow2 75.6%

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

      2. unpow2 75.6%

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

      3. swap-sqr 98.6%

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

      4. unpow2 98.6%

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

   8. Simplified 98.6%

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

      1. unpow2 98.6%

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

   10. Applied egg-rr 98.6%

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

   if -1.05000000000000004 < v < 7.2000000000000002e-19

   1. Initial program 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in v around 0 77.5%

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

      1. *-commutative 77.5%

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

      2. unpow2 77.5%

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

      3. unpow2 77.5%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. remove-double-div 99.8%

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

      3. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.8%

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

      3. associate-*l* 99.8%

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

   10. Applied egg-rr 99.8%

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

   if 7.2000000000000002e-19 < v

   1. Initial program 75.9%

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

   3. Simplified 95.7%

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

      1. associate-*r* 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. times-frac 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 83.6%

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

      1. unpow2 83.6%

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

      2. unpow2 83.6%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   8. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. remove-double-div 99.8%

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

      3. /-rgt-identity 99.8%

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

      4. associate-/l* 99.8%

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

      5. frac-times 95.6%

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

      6. *-un-lft-identity 95.6%

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

   10. Applied egg-rr 95.6%

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

       1. div-inv 95.7%

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

       2. frac-times 95.7%

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

       3. metadata-eval 95.7%

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

       4. remove-double-div 95.7%

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

       5. *-commutative 95.7%

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

       6. associate-*r* 99.8%

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

       7. associate-*r* 99.8%

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

   12. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.05:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 7.2 \cdot 10^{-19}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{r \cdot \left(w \cdot -0.375\right)}{\frac{\frac{1}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -2:\\ \;\;\;\;t_0 + \left(-1.5 - 0.25 \cdot \frac{r \cdot w}{\frac{1}{r \cdot w}}\right)\\ \mathbf{elif}\;v \leq 7.2 \cdot 10^{-19}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{r \cdot \left(w \cdot -0.375\right)}{\frac{\frac{1}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (<= v -2.0)
     (+ t_0 (- -1.5 (* 0.25 (/ (* r w) (/ 1.0 (* r w))))))
     (if (<= v 7.2e-19)
       (+ -1.5 (+ (/ 2.0 (* r r)) (/ (* r (* w -0.375)) (/ (/ 1.0 r) w))))
       (+ t_0 (- -1.5 (* 0.25 (* w (* r (* r w))))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if (v <= -2.0) {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) / (1.0 / (r * w)))));
	} else if (v <= 7.2e-19) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * -0.375)) / ((1.0 / r) / w)));
	} else {
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (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 <= (-2.0d0)) then
        tmp = t_0 + ((-1.5d0) - (0.25d0 * ((r * w) / (1.0d0 / (r * w)))))
    else if (v <= 7.2d-19) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((r * (w * (-0.375d0))) / ((1.0d0 / r) / w)))
    else
        tmp = t_0 + ((-1.5d0) - (0.25d0 * (w * (r * (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 <= -2.0) {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) / (1.0 / (r * w)))));
	} else if (v <= 7.2e-19) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * -0.375)) / ((1.0 / r) / w)));
	} else {
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	tmp = 0
	if v <= -2.0:
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) / (1.0 / (r * w)))))
	elif v <= 7.2e-19:
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * -0.375)) / ((1.0 / r) / w)))
	else:
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (r * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if (v <= -2.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * Float64(Float64(r * w) / Float64(1.0 / Float64(r * w))))));
	elseif (v <= 7.2e-19)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(r * Float64(w * -0.375)) / Float64(Float64(1.0 / r) / w))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * Float64(w * Float64(r * 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 <= -2.0)
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) / (1.0 / (r * w)))));
	elseif (v <= 7.2e-19)
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * (w * -0.375)) / ((1.0 / r) / w)));
	else
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (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[LessEqual[v, -2.0], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[(N[(r * w), $MachinePrecision] / N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 7.2e-19], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2

   1. Initial program 72.7%

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

   3. Simplified 97.0%

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

      1. associate-*r* 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. times-frac 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in v around inf 75.6%

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

      1. unpow2 75.6%

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

      2. unpow2 75.6%

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

      3. swap-sqr 98.6%

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

      4. unpow2 98.6%

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

   8. Simplified 98.6%

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

      1. unpow2 81.1%

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

      2. remove-double-div 81.1%

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

      3. un-div-inv 81.1%

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

   10. Applied egg-rr 98.7%

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

   if -2 < v < 7.2000000000000002e-19

   1. Initial program 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in v around 0 77.5%

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

      1. *-commutative 77.5%

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

      2. unpow2 77.5%

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

      3. unpow2 77.5%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. remove-double-div 99.8%

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

      3. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.8%

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

      3. associate-*l* 99.8%

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

   10. Applied egg-rr 99.8%

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

   if 7.2000000000000002e-19 < v

   1. Initial program 75.9%

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

   3. Simplified 95.7%

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

      1. associate-*r* 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. times-frac 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 83.6%

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

      1. unpow2 83.6%

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

      2. unpow2 83.6%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   8. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. remove-double-div 99.8%

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

      3. /-rgt-identity 99.8%

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

      4. associate-/l* 99.8%

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

      5. frac-times 95.6%

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

      6. *-un-lft-identity 95.6%

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

   10. Applied egg-rr 95.6%

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

       1. div-inv 95.7%

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

       2. frac-times 95.7%

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

       3. metadata-eval 95.7%

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

       4. remove-double-div 95.7%

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

       5. *-commutative 95.7%

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

       6. associate-*r* 99.8%

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

       7. associate-*r* 99.8%

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

   12. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \frac{r \cdot w}{\frac{1}{r \cdot w}}\right)\\ \mathbf{elif}\;v \leq 7.2 \cdot 10^{-19}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{r \cdot \left(w \cdot -0.375\right)}{\frac{\frac{1}{r}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.12 \lor \neg \left(v \leq 7.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -1.12) (not (<= v 7.2e-19)))
   (+ (/ (/ 2.0 r) r) (- -1.5 (* 0.25 (* (* r w) (* r w)))))
   (+ -1.5 (+ (/ 2.0 (* r r)) (* (* r w) (* w (* r -0.375)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -1.12) || !(v <= 7.2e-19)) {
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * ((r * w) * (r * w))));
	} else {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * w) * (w * (r * -0.375))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if ((v <= (-1.12d0)) .or. (.not. (v <= 7.2d-19))) then
        tmp = ((2.0d0 / r) / r) + ((-1.5d0) - (0.25d0 * ((r * w) * (r * w))))
    else
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((r * w) * (w * (r * (-0.375d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -1.12) || !(v <= 7.2e-19)) {
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * ((r * w) * (r * w))));
	} else {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * w) * (w * (r * -0.375))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -1.12) or not (v <= 7.2e-19):
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * ((r * w) * (r * w))))
	else:
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * w) * (w * (r * -0.375))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -1.12) || !(v <= 7.2e-19))
		tmp = Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(0.25 * Float64(Float64(r * w) * Float64(r * w)))));
	else
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(r * w) * Float64(w * Float64(r * -0.375)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -1.12) || ~((v <= 7.2e-19)))
		tmp = ((2.0 / r) / r) + (-1.5 - (0.25 * ((r * w) * (r * w))));
	else
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * w) * (w * (r * -0.375))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -1.12], N[Not[LessEqual[v, 7.2e-19]], $MachinePrecision]], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.1200000000000001 or 7.2000000000000002e-19 < v

   1. Initial program 74.3%

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

   3. Simplified 96.3%

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

      1. associate-*r* 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. times-frac 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 79.7%

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

      1. unpow2 79.7%

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

      2. unpow2 79.7%

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

      3. swap-sqr 99.2%

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

      4. unpow2 99.2%

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

   8. Simplified 99.2%

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

      1. unpow2 99.2%

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

   10. Applied egg-rr 99.2%

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

   if -1.1200000000000001 < v < 7.2000000000000002e-19

   1. Initial program 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in v around 0 77.5%

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

      1. *-commutative 77.5%

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

      2. unpow2 77.5%

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

      3. unpow2 77.5%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. remove-double-div 99.8%

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

      3. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.8%

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

      3. associate-*l* 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. div-inv 99.8%

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

       2. associate-*r* 99.7%

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

       3. *-commutative 99.7%

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

       4. associate-*l* 99.8%

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

       5. associate-/r* 99.8%

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

       6. remove-double-div 99.8%

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

   12. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.12 \lor \neg \left(v \leq 7.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]

Alternative 7: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -1.25:\\ \;\;\;\;t_0 + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 7.2 \cdot 10^{-19}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (<= v -1.25)
     (+ t_0 (- -1.5 (* 0.25 (* (* r w) (* r w)))))
     (if (<= v 7.2e-19)
       (+ -1.5 (+ (/ 2.0 (* r r)) (* (* r w) (* w (* r -0.375)))))
       (+ t_0 (- -1.5 (* 0.25 (* w (* r (* r w))))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if (v <= -1.25) {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	} else if (v <= 7.2e-19) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * w) * (w * (r * -0.375))));
	} else {
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (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 <= (-1.25d0)) then
        tmp = t_0 + ((-1.5d0) - (0.25d0 * ((r * w) * (r * w))))
    else if (v <= 7.2d-19) then
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + ((r * w) * (w * (r * (-0.375d0)))))
    else
        tmp = t_0 + ((-1.5d0) - (0.25d0 * (w * (r * (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 <= -1.25) {
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	} else if (v <= 7.2e-19) {
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * w) * (w * (r * -0.375))));
	} else {
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	tmp = 0
	if v <= -1.25:
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))))
	elif v <= 7.2e-19:
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * w) * (w * (r * -0.375))))
	else:
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (r * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if (v <= -1.25)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * Float64(Float64(r * w) * Float64(r * w)))));
	elseif (v <= 7.2e-19)
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(r * w) * Float64(w * Float64(r * -0.375)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * Float64(w * Float64(r * 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 <= -1.25)
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) * (r * w))));
	elseif (v <= 7.2e-19)
		tmp = -1.5 + ((2.0 / (r * r)) + ((r * w) * (w * (r * -0.375))));
	else
		tmp = t_0 + (-1.5 - (0.25 * (w * (r * (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[LessEqual[v, -1.25], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 7.2e-19], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.25

   1. Initial program 72.7%

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

   3. Simplified 97.0%

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

      1. associate-*r* 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. times-frac 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in v around inf 75.6%

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

      1. unpow2 75.6%

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

      2. unpow2 75.6%

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

      3. swap-sqr 98.6%

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

      4. unpow2 98.6%

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

   8. Simplified 98.6%

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

      1. unpow2 98.6%

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

   10. Applied egg-rr 98.6%

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

   if -1.25 < v < 7.2000000000000002e-19

   1. Initial program 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in v around 0 77.5%

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

      1. *-commutative 77.5%

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

      2. unpow2 77.5%

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

      3. unpow2 77.5%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. remove-double-div 99.8%

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

      3. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.8%

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

      3. associate-*l* 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. div-inv 99.8%

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

       2. associate-*r* 99.7%

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

       3. *-commutative 99.7%

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

       4. associate-*l* 99.8%

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

       5. associate-/r* 99.8%

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

       6. remove-double-div 99.8%

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

   12. Applied egg-rr 99.8%

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

   if 7.2000000000000002e-19 < v

   1. Initial program 75.9%

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

   3. Simplified 95.7%

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

      1. associate-*r* 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. times-frac 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 83.6%

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

      1. unpow2 83.6%

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

      2. unpow2 83.6%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   8. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. remove-double-div 99.8%

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

      3. /-rgt-identity 99.8%

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

      4. associate-/l* 99.8%

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

      5. frac-times 95.6%

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

      6. *-un-lft-identity 95.6%

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

   10. Applied egg-rr 95.6%

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

       1. div-inv 95.7%

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

       2. frac-times 95.7%

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

       3. metadata-eval 95.7%

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

       4. remove-double-div 95.7%

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

       5. *-commutative 95.7%

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

       6. associate-*r* 99.8%

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

       7. associate-*r* 99.8%

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

   12. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.25:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 7.2 \cdot 10^{-19}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + r \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot -0.375\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2e-137)
   (+ (/ (/ 2.0 r) r) -1.5)
   (+ -1.5 (+ (/ 2.0 (* r r)) (* r (* (* r w) (* w -0.375)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-137) {
		tmp = ((2.0 / r) / r) + -1.5;
	} else {
		tmp = -1.5 + ((2.0 / (r * r)) + (r * ((r * w) * (w * -0.375))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2d-137) then
        tmp = ((2.0d0 / r) / r) + (-1.5d0)
    else
        tmp = (-1.5d0) + ((2.0d0 / (r * r)) + (r * ((r * w) * (w * (-0.375d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-137) {
		tmp = ((2.0 / r) / r) + -1.5;
	} else {
		tmp = -1.5 + ((2.0 / (r * r)) + (r * ((r * w) * (w * -0.375))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2e-137:
		tmp = ((2.0 / r) / r) + -1.5
	else:
		tmp = -1.5 + ((2.0 / (r * r)) + (r * ((r * w) * (w * -0.375))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2e-137)
		tmp = Float64(Float64(Float64(2.0 / r) / r) + -1.5);
	else
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r * r)) + Float64(r * Float64(Float64(r * w) * Float64(w * -0.375)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2e-137)
		tmp = ((2.0 / r) / r) + -1.5;
	else
		tmp = -1.5 + ((2.0 / (r * r)) + (r * ((r * w) * (w * -0.375))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2e-137], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + -1.5), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(r * N[(N[(r * w), $MachinePrecision] * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.99999999999999996e-137

   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 94.5%

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

      1. associate-*l/ 88.7%

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

      2. associate-/l* 94.5%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in r around 0 61.6%

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

   if 1.99999999999999996e-137 < r

   1. Initial program 84.1%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in v around 0 74.2%

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

      1. *-commutative 74.2%

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

      2. unpow2 74.2%

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

      3. unpow2 74.2%

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

      4. swap-sqr 87.7%

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

      5. unpow2 87.7%

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

   6. Simplified 87.7%

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

      1. unpow2 87.7%

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

      2. remove-double-div 87.7%

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

      3. un-div-inv 87.7%

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

   8. Applied egg-rr 87.7%

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

      1. associate-*l/ 87.7%

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

      2. associate-/r* 87.7%

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

      3. associate-*l* 87.7%

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

   10. Applied egg-rr 87.7%

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

       1. div-inv 87.7%

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

       2. associate-/r* 87.7%

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

       3. remove-double-div 87.7%

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

       4. associate-*l* 86.7%

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

       5. *-commutative 86.7%

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

   12. Applied egg-rr 86.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + r \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot -0.375\right)\right)\right)\\ \end{array} \]

Alternative 9: 91.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Simplified 87.5%

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

4. Taylor expanded in v around 0 73.4%

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

   1. *-commutative 73.4%

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

   2. unpow2 73.4%

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

   3. unpow2 73.4%

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

   4. swap-sqr 91.1%

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

   5. unpow2 91.1%

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

6. Simplified 91.1%

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

   1. unpow2 91.1%

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

   2. remove-double-div 91.1%

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

   3. un-div-inv 91.1%

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

8. Applied egg-rr 91.1%

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

   1. associate-*l/ 91.1%

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

   2. associate-/r* 91.1%

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

   3. associate-*l* 91.1%

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

10. Applied egg-rr 91.1%

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

    1. div-inv 91.1%

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

    2. *-commutative 91.1%

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

    3. associate-/r* 91.1%

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

    4. remove-double-div 91.1%

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

    5. *-commutative 91.1%

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

    6. associate-*l* 91.1%

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

    7. *-commutative 91.1%

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

    8. associate-*r* 91.1%

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

    9. *-commutative 91.1%

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

    10. associate-*l* 87.6%

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

    11. *-commutative 87.6%

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

    12. associate-*l* 87.6%

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

    13. *-commutative 87.6%

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

    14. associate-*l* 90.2%

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

12. Applied egg-rr 90.2%

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

13. Final simplification 90.2%

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

Alternative 10: 93.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Simplified 87.5%

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

4. Taylor expanded in v around 0 73.4%

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

   1. *-commutative 73.4%

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

   2. unpow2 73.4%

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

   3. unpow2 73.4%

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

   4. swap-sqr 91.1%

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

   5. unpow2 91.1%

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

6. Simplified 91.1%

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

   1. unpow2 91.1%

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

   2. remove-double-div 91.1%

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

   3. un-div-inv 91.1%

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

8. Applied egg-rr 91.1%

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

   1. associate-*l/ 91.1%

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

   2. associate-/r* 91.1%

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

   3. associate-*l* 91.1%

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

10. Applied egg-rr 91.1%

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

    1. div-inv 91.1%

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

    2. associate-*r* 91.1%

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

    3. *-commutative 91.1%

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

    4. associate-*l* 91.1%

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

    5. associate-/r* 91.1%

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

    6. remove-double-div 91.1%

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

12. Applied egg-rr 91.1%

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

13. Final simplification 91.1%

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

Alternative 11: 13.7% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Simplified 96.0%

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

   1. associate-*l/ 88.6%

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

   2. associate-/l* 96.0%

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

   3. associate-*r* 99.7%

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

   4. pow2 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in r around 0 56.5%

   \[\leadsto \frac{\frac{2}{r}}{r} + \color{blue}{-1.5} \]
7. Step-by-step derivation

   1. div-inv 56.5%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{r}}}{r} + -1.5 \]

   2. add-sqr-sqrt 28.3%

      \[\leadsto \frac{2 \cdot \frac{1}{r}}{\color{blue}{\sqrt{r} \cdot \sqrt{r}}} + -1.5 \]

   3. sqrt-prod 33.7%

      \[\leadsto \frac{2 \cdot \frac{1}{r}}{\color{blue}{\sqrt{r \cdot r}}} + -1.5 \]

   4. sqr-neg 33.7%

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

   5. sqrt-unprod 5.4%

      \[\leadsto \frac{2 \cdot \frac{1}{r}}{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}} + -1.5 \]

   6. add-sqr-sqrt 13.7%

      \[\leadsto \frac{2 \cdot \frac{1}{r}}{\color{blue}{-r}} + -1.5 \]

   7. neg-mul-1 13.7%

      \[\leadsto \frac{2 \cdot \frac{1}{r}}{\color{blue}{-1 \cdot r}} + -1.5 \]

   8. metadata-eval 13.7%

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

   9. times-frac 13.7%

      \[\leadsto \color{blue}{\frac{2}{-1} \cdot \frac{\frac{1}{r}}{r}} + -1.5 \]

   10. metadata-eval 13.7%

       \[\leadsto \frac{2}{\color{blue}{-1}} \cdot \frac{\frac{1}{r}}{r} + -1.5 \]

   11. metadata-eval 13.7%

       \[\leadsto \color{blue}{-2} \cdot \frac{\frac{1}{r}}{r} + -1.5 \]

8. Applied egg-rr 13.7%

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

   1. associate-*r/ 13.7%

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{1}{r}}{r}} + -1.5 \]

   2. un-div-inv 13.7%

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

10. Applied egg-rr 13.7%

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

11. Final simplification 13.7%

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

Alternative 12: 57.6% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Simplified 96.0%

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

   1. associate-*l/ 88.6%

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

   2. associate-/l* 96.0%

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

   3. associate-*r* 99.7%

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

   4. pow2 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in r around 0 56.5%

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

7. Final simplification 56.5%

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

Reproduce

herbie shell --seed 2023331 
(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))