Rosa's TurbineBenchmark

Percentage Accurate: 84.7% → 99.8%

Time: 13.8s

Alternatives: 9

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: 84.7% 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

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

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 / Float64(r * r)) + -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 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] - 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]

Derivation

1. Initial program 80.4%

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

3. Simplified 96.1%

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

   1. associate-*r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

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 / Float64(r * r)) + -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 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] - 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]

Derivation

1. Initial program 80.4%

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

3. Simplified 96.1%

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

   1. associate-*r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

   1. /-rgt-identity 99.8%

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

   2. clear-num 99.8%

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

   3. un-div-inv 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (-1.5d0) + ((2.0d0 / (r * r)) + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * ((r * w) ** 2.0d0)))
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)) * Math.pow((r * w), 2.0)));
}

Python

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

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(r * w) ^ 2.0))))
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) ^ 2.0)));
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[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

3. Simplified 83.7%

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

4. Taylor expanded in r around 0 76.1%

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

   1. unpow2 76.1%

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

   2. unpow2 76.1%

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

   3. swap-sqr 99.8%

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

   4. unpow2 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 6 \cdot 10^{+176}:\\ \;\;\;\;-1.5 + \left(t_0 + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + -1.5\right) - \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 6e+176)
     (+
      -1.5
      (+ t_0 (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* w (* r (* r w))))))
     (- (+ t_0 -1.5) (/ (* r (* w (* r w))) 4.0)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6e+176) {
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (w * (r * (r * w)))));
	} else {
		tmp = (t_0 + -1.5) - ((r * (w * (r * w))) / 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 6d+176) then
        tmp = (-1.5d0) + (t_0 + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (w * (r * (r * w)))))
    else
        tmp = (t_0 + (-1.5d0)) - ((r * (w * (r * w))) / 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6e+176) {
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (w * (r * (r * w)))));
	} else {
		tmp = (t_0 + -1.5) - ((r * (w * (r * w))) / 4.0);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 6e+176:
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (w * (r * (r * w)))))
	else:
		tmp = (t_0 + -1.5) - ((r * (w * (r * w))) / 4.0)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 6e+176)
		tmp = Float64(-1.5 + Float64(t_0 + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(w * Float64(r * Float64(r * w))))));
	else
		tmp = Float64(Float64(t_0 + -1.5) - Float64(Float64(r * Float64(w * Float64(r * w))) / 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 6e+176)
		tmp = -1.5 + (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (w * (r * (r * w)))));
	else
		tmp = (t_0 + -1.5) - ((r * (w * (r * w))) / 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 6e+176], N[(-1.5 + N[(t$95$0 + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + -1.5), $MachinePrecision] - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 6e176

   1. Initial program 81.2%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in r around 0 79.8%

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

      1. unpow2 79.8%

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

      2. unpow2 79.8%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 94.0%

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

      2. associate-*r* 93.9%

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

   8. Applied egg-rr 99.0%

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

   if 6e176 < r

   1. Initial program 74.4%

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

   3. Simplified 99.9%

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

      1. associate-*l/ 87.3%

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

      2. associate-/l* 99.9%

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

      3. associate-*r* 99.7%

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

      4. pow2 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in v around inf 91.8%

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

      1. unpow2 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-*r* 75.2%

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

   8. Applied egg-rr 91.9%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(\left(r \cdot w\right) \cdot w\right) \cdot r}}{4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6 \cdot 10^{+176}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{4}\\ \end{array} \]

Alternative 5: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq -2.8 \cdot 10^{-41} \lor \neg \left(v \leq 1.4 \cdot 10^{-67}\right):\\ \;\;\;\;t_0 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{4}\\ \mathbf{else}:\\ \;\;\;\;t_0 - 0.375 \cdot \left(w \cdot \frac{r \cdot w}{\frac{1}{r}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5)))
   (if (or (<= v -2.8e-41) (not (<= v 1.4e-67)))
     (- t_0 (/ (* (* r w) (* r w)) 4.0))
     (- t_0 (* 0.375 (* w (/ (* r w) (/ 1.0 r))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if ((v <= -2.8e-41) || !(v <= 1.4e-67)) {
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	} else {
		tmp = t_0 - (0.375 * (w * ((r * w) / (1.0 / r))));
	}
	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)) + (-1.5d0)
    if ((v <= (-2.8d-41)) .or. (.not. (v <= 1.4d-67))) then
        tmp = t_0 - (((r * w) * (r * w)) / 4.0d0)
    else
        tmp = t_0 - (0.375d0 * (w * ((r * w) / (1.0d0 / r))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if ((v <= -2.8e-41) || !(v <= 1.4e-67)) {
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	} else {
		tmp = t_0 - (0.375 * (w * ((r * w) / (1.0 / r))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if (v <= -2.8e-41) or not (v <= 1.4e-67):
		tmp = t_0 - (((r * w) * (r * w)) / 4.0)
	else:
		tmp = t_0 - (0.375 * (w * ((r * w) / (1.0 / r))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if ((v <= -2.8e-41) || !(v <= 1.4e-67))
		tmp = Float64(t_0 - Float64(Float64(Float64(r * w) * Float64(r * w)) / 4.0));
	else
		tmp = Float64(t_0 - Float64(0.375 * Float64(w * Float64(Float64(r * w) / Float64(1.0 / r)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if ((v <= -2.8e-41) || ~((v <= 1.4e-67)))
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	else
		tmp = t_0 - (0.375 * (w * ((r * w) / (1.0 / r))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[Or[LessEqual[v, -2.8e-41], N[Not[LessEqual[v, 1.4e-67]], $MachinePrecision]], N[(t$95$0 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(0.375 * N[(w * N[(N[(r * w), $MachinePrecision] / N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.8000000000000002e-41 or 1.40000000000000005e-67 < v

   1. Initial program 78.5%

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

   3. Simplified 98.0%

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

      1. associate-*l/ 88.5%

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

      2. associate-/l* 98.0%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 99.0%

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

      1. unpow2 99.0%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}{4} \]

   8. Applied egg-rr 99.0%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}{4} \]

   if -2.8000000000000002e-41 < v < 1.40000000000000005e-67

   1. Initial program 83.5%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in v around 0 75.3%

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

      1. *-commutative 75.3%

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

      2. unpow2 75.3%

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

      3. unpow2 75.3%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. associate-*r* 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. *-commutative 97.9%

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

      2. /-rgt-identity 97.9%

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

      3. clear-num 97.9%

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

      4. div-inv 97.9%

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

      5. clear-num 97.9%

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

      6. div-inv 97.9%

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

      7. div-inv 97.8%

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

      8. associate-/r* 97.9%

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

      9. div-inv 97.9%

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

      10. clear-num 97.9%

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

      11. /-rgt-identity 97.9%

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

   10. Applied egg-rr 97.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.8 \cdot 10^{-41} \lor \neg \left(v \leq 1.4 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - 0.375 \cdot \left(w \cdot \frac{r \cdot w}{\frac{1}{r}}\right)\\ \end{array} \]

Alternative 6: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq -4 \cdot 10^{+30} \lor \neg \left(v \leq 5 \cdot 10^{-67}\right):\\ \;\;\;\;t_0 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{4}\\ \mathbf{else}:\\ \;\;\;\;t_0 - 0.375 \cdot \frac{r \cdot w}{\frac{\frac{1}{w}}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5)))
   (if (or (<= v -4e+30) (not (<= v 5e-67)))
     (- t_0 (/ (* (* r w) (* r w)) 4.0))
     (- t_0 (* 0.375 (/ (* r w) (/ (/ 1.0 w) r)))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if ((v <= -4e+30) || !(v <= 5e-67)) {
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	} else {
		tmp = t_0 - (0.375 * ((r * w) / ((1.0 / w) / r)));
	}
	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)) + (-1.5d0)
    if ((v <= (-4d+30)) .or. (.not. (v <= 5d-67))) then
        tmp = t_0 - (((r * w) * (r * w)) / 4.0d0)
    else
        tmp = t_0 - (0.375d0 * ((r * w) / ((1.0d0 / w) / r)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if ((v <= -4e+30) || !(v <= 5e-67)) {
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	} else {
		tmp = t_0 - (0.375 * ((r * w) / ((1.0 / w) / r)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if (v <= -4e+30) or not (v <= 5e-67):
		tmp = t_0 - (((r * w) * (r * w)) / 4.0)
	else:
		tmp = t_0 - (0.375 * ((r * w) / ((1.0 / w) / r)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if ((v <= -4e+30) || !(v <= 5e-67))
		tmp = Float64(t_0 - Float64(Float64(Float64(r * w) * Float64(r * w)) / 4.0));
	else
		tmp = Float64(t_0 - Float64(0.375 * Float64(Float64(r * w) / Float64(Float64(1.0 / w) / r))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if ((v <= -4e+30) || ~((v <= 5e-67)))
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	else
		tmp = t_0 - (0.375 * ((r * w) / ((1.0 / w) / r)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[Or[LessEqual[v, -4e+30], N[Not[LessEqual[v, 5e-67]], $MachinePrecision]], N[(t$95$0 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(0.375 * N[(N[(r * w), $MachinePrecision] / N[(N[(1.0 / w), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -4.0000000000000001e30 or 4.9999999999999999e-67 < v

   1. Initial program 77.2%

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

   3. Simplified 97.8%

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

      1. associate-*l/ 87.1%

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

      2. associate-/l* 97.8%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 99.4%

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

      1. unpow2 99.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}{4} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}{4} \]

   if -4.0000000000000001e30 < v < 4.9999999999999999e-67

   1. Initial program 84.3%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in v around 0 75.8%

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

      1. *-commutative 75.8%

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

      2. unpow2 75.8%

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

      3. unpow2 75.8%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. remove-double-div 99.8%

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

      3. un-div-inv 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-/r* 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{+30} \lor \neg \left(v \leq 5 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - 0.375 \cdot \frac{r \cdot w}{\frac{\frac{1}{w}}{r}}\\ \end{array} \]

Alternative 7: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq -2.9 \cdot 10^{+30} \lor \neg \left(v \leq 5 \cdot 10^{-67}\right):\\ \;\;\;\;t_0 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{4}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{r \cdot w}{\frac{\frac{\frac{1}{r}}{w}}{0.375}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5)))
   (if (or (<= v -2.9e+30) (not (<= v 5e-67)))
     (- t_0 (/ (* (* r w) (* r w)) 4.0))
     (- t_0 (/ (* r w) (/ (/ (/ 1.0 r) w) 0.375))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if ((v <= -2.9e+30) || !(v <= 5e-67)) {
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	} else {
		tmp = t_0 - ((r * w) / (((1.0 / r) / w) / 0.375));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / (r * r)) + (-1.5d0)
    if ((v <= (-2.9d+30)) .or. (.not. (v <= 5d-67))) then
        tmp = t_0 - (((r * w) * (r * w)) / 4.0d0)
    else
        tmp = t_0 - ((r * w) / (((1.0d0 / r) / w) / 0.375d0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if ((v <= -2.9e+30) || !(v <= 5e-67)) {
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	} else {
		tmp = t_0 - ((r * w) / (((1.0 / r) / w) / 0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if (v <= -2.9e+30) or not (v <= 5e-67):
		tmp = t_0 - (((r * w) * (r * w)) / 4.0)
	else:
		tmp = t_0 - ((r * w) / (((1.0 / r) / w) / 0.375))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if ((v <= -2.9e+30) || !(v <= 5e-67))
		tmp = Float64(t_0 - Float64(Float64(Float64(r * w) * Float64(r * w)) / 4.0));
	else
		tmp = Float64(t_0 - Float64(Float64(r * w) / Float64(Float64(Float64(1.0 / r) / w) / 0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if ((v <= -2.9e+30) || ~((v <= 5e-67)))
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	else
		tmp = t_0 - ((r * w) / (((1.0 / r) / w) / 0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[Or[LessEqual[v, -2.9e+30], N[Not[LessEqual[v, 5e-67]], $MachinePrecision]], N[(t$95$0 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(r * w), $MachinePrecision] / N[(N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.8999999999999998e30 or 4.9999999999999999e-67 < v

   1. Initial program 77.2%

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

   3. Simplified 97.8%

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

      1. associate-*l/ 87.1%

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

      2. associate-/l* 97.8%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 99.4%

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

      1. unpow2 99.4%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}{4} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}{4} \]

   if -2.8999999999999998e30 < v < 4.9999999999999999e-67

   1. Initial program 84.3%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in v around 0 75.8%

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

      1. *-commutative 75.8%

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

      2. unpow2 75.8%

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

      3. unpow2 75.8%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*r* 93.9%

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

   8. Applied egg-rr 93.9%

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

      1. associate-*l* 99.8%

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

      2. *-commutative 99.8%

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

      3. remove-double-div 99.8%

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

      4. associate-/l/ 99.8%

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

      5. div-inv 99.8%

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

      6. associate-*l/ 99.8%

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

      7. associate-/l* 99.9%

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

      8. associate-/l/ 99.8%

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

      9. associate-/r* 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.9 \cdot 10^{+30} \lor \neg \left(v \leq 5 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot w}{\frac{\frac{\frac{1}{r}}{w}}{0.375}}\\ \end{array} \]

Alternative 8: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq -7.5 \cdot 10^{-43} \lor \neg \left(v \leq 6.6 \cdot 10^{-67}\right):\\ \;\;\;\;t_0 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{4}\\ \mathbf{else}:\\ \;\;\;\;t_0 - 0.375 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5)))
   (if (or (<= v -7.5e-43) (not (<= v 6.6e-67)))
     (- t_0 (/ (* (* r w) (* r w)) 4.0))
     (- t_0 (* 0.375 (* w (* r (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if ((v <= -7.5e-43) || !(v <= 6.6e-67)) {
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	} else {
		tmp = t_0 - (0.375 * (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)) + (-1.5d0)
    if ((v <= (-7.5d-43)) .or. (.not. (v <= 6.6d-67))) then
        tmp = t_0 - (((r * w) * (r * w)) / 4.0d0)
    else
        tmp = t_0 - (0.375d0 * (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)) + -1.5;
	double tmp;
	if ((v <= -7.5e-43) || !(v <= 6.6e-67)) {
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	} else {
		tmp = t_0 - (0.375 * (w * (r * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if (v <= -7.5e-43) or not (v <= 6.6e-67):
		tmp = t_0 - (((r * w) * (r * w)) / 4.0)
	else:
		tmp = t_0 - (0.375 * (w * (r * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if ((v <= -7.5e-43) || !(v <= 6.6e-67))
		tmp = Float64(t_0 - Float64(Float64(Float64(r * w) * Float64(r * w)) / 4.0));
	else
		tmp = Float64(t_0 - Float64(0.375 * 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)) + -1.5;
	tmp = 0.0;
	if ((v <= -7.5e-43) || ~((v <= 6.6e-67)))
		tmp = t_0 - (((r * w) * (r * w)) / 4.0);
	else
		tmp = t_0 - (0.375 * (w * (r * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[Or[LessEqual[v, -7.5e-43], N[Not[LessEqual[v, 6.6e-67]], $MachinePrecision]], N[(t$95$0 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(0.375 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -7.50000000000000068e-43 or 6.6000000000000003e-67 < v

   1. Initial program 78.5%

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

   3. Simplified 98.0%

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

      1. associate-*l/ 88.5%

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

      2. associate-/l* 98.0%

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

      3. associate-*r* 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 99.0%

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

      1. unpow2 99.0%

         \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}{4} \]

   8. Applied egg-rr 99.0%

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}{4} \]

   if -7.50000000000000068e-43 < v < 6.6000000000000003e-67

   1. Initial program 83.5%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in v around 0 75.3%

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

      1. *-commutative 75.3%

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

      2. unpow2 75.3%

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

      3. unpow2 75.3%

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

      4. swap-sqr 99.8%

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

      5. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. associate-*r* 97.9%

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

   8. Applied egg-rr 97.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -7.5 \cdot 10^{-43} \lor \neg \left(v \leq 6.6 \cdot 10^{-67}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - 0.375 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 9: 91.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

3. Simplified 96.1%

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

4. Taylor expanded in v around 0 73.0%

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

   1. *-commutative 73.0%

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

   2. unpow2 73.0%

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

   3. unpow2 73.0%

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

   4. swap-sqr 91.8%

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

   5. unpow2 91.8%

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

6. Simplified 91.8%

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

   1. unpow2 91.8%

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

   2. associate-*r* 90.4%

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

8. Applied egg-rr 90.4%

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

9. Final simplification 90.4%

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

Reproduce

herbie shell --seed 2023320 
(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))